AN INTRODUCTION TO
BLACK HOLES, INFORMATION,
AND THE STRING THEORY
REVOLUTION
The Holographic Universe
Leonard Susskind∗
James Lindesay†
∗ Permanent
address, Department of Physics, Stanford University, Stanford, CA 943054060
† Permanent address, Department of Physics, Howard University, Washington, DC 20059
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AN INTRODUCTION TO BLACK HOLES, INFORMATION AND THE STRING
THEORY REVOLUTION
The Holographic Universe
Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd.
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Preface
It is now almost a century since the year 1905, in which the principle
of relativity and the hypothesis of the quantum of radiation were introduced. It has taken most of that time to synthesize the two into the modern
quantum theory of fields and the standard model of particle phenomena.
Although there is undoubtably more to be learned both theoretically and
experimentally, it seems likely that we know most of the basic principles
which follow from combining the special theory of relativity with quantum
mechanics. It is unlikely that a major revolution will spring from this soil.
By contrast, in the 80 years that we have had the general theory of relativity, nothing comparable has been learned about the quantum theory of
gravitation. The methods that were invented to quantize electrodynamics,
which were so successfully generalized to build the standard model, prove
wholly inadequate when applied to gravitation. The subject is riddled with
paradox and contradiction. One has the distinct impression that we are
thinking about the things in the wrong way. The paradigm of relativistic
quantum field theory almost certainly has to be replaced.
How then are we to go about finding the right replacement? It seems
very unlikely that the usual incremental increase of knowledge from a combination of theory and experiment will ever get us where we want to go,
that is, to the Planck scale. Under this circumstance our best hope is an
examination of fundamental principles, paradoxes and contradictions, and
the study of gedanken experiments. Such strategy has worked in the past.
The earliest origins of quantum mechanics were not experimental atomic
physics, radioactivity, or spectral lines. The puzzle which started the whole
thing was a contradiction between the principles of statistical thermodynamics and the field concept of Faraday and Maxwell. How was it possible,
Planck asked, for the infinite collection of radiation oscillators to have a
finite specific heat?
vii
viii
Black Holes, Information, and the String Theory Revolution
In the case of special relativity it was again a conceptual contradiction
and a gedanken experiment which opened the way. According to Einstein,
at the age of 15 he formulated the following paradox: suppose an observer
moved along with a light beam and observed it. The electromagnetic field
would be seen as a static, spatially varying field. But no such solution
to Maxwell’s equations exists. By this simple means a contradiction was
exposed between the symmetries of Newton’s and Galileo’s mechanics and
those of Maxwell’s electrodynamics.
The development of the general theory from the principle of equivalence and the man-in-the-elevator gedanken experiment is also a matter of
historical fact. In each of these cases the consistency of readily observed
properties of nature which had been known for many years required revolutionary paradigm shifts.
What known properties of nature should we look to, and which paradox
is best suited to our present purposes? Certainly the most important facts
are the success of the general theory in describing gravity and of quantum
mechanics in describing the microscopic world. Furthermore, the two theories appear to lead to a serious clash that once again involves statistical
thermodynamics in an essential way. The paradox was discovered by Jacob Bekenstein and turned into a serious crisis by Stephen Hawking. By
an analysis of gedanken experiments, Bekenstein realized that if the second law of thermodynamics was not to be violated in the presence of a
black hole, the black hole must possess an intrinsic entropy. This in itself
is a source of paradox. How and why a classical solution of field equations
should be endowed with thermodynamical attributes has remained obscure
since Bekenstein’s discovery in 1972.
Hawking added to the puzzle when he discovered that a black hole will
radiate away its energy in the form of Planckian black body radiation.
Eventually the black hole must completely evaporate. Hawking then raised
the question of what becomes of the quantum correlations between matter
outside the black hole and matter that disappears behind the horizon. As
long as the black hole is present, one can do the bookkeeping so that it is the
black hole itself which is correlated to the matter outside. But eventually
the black hole will evaporate. Hawking then made arguments that there is
no way, consistent with causality, for the correlations to be carried by the
outgoing evaporation products. Thus, according to Hawking, the existence
of black holes inevitably causes a loss of quantum coherence and breakdown
of one of the basic principles of quantum mechanics – the evolution of
pure states to pure states. For two decades this contradiction between
Preface
ix
the principles of general relativity and quantum mechanics has so puzzled
theorists that many now see it as a serious crisis.
Hawking and much of the traditional relativity community have been of
the opinion that the correct resolution of the paradox is simply that quantum coherence is lost during black hole evaporation. From an operational
viewpoint this would mean that the standard rules of quantum mechanics
would not apply to processes involving black holes. Hawking further argued that once the loss of quantum coherence is permitted in black hole
evaporation, it becomes compulsory in all processes involving the Planck
scale. The world would behave as if it were in a noisy environment which
continuously leads to a loss of coherence. The trouble with this is that there
is no known way to destroy coherence without, at the same time violating
energy conservation by heating the world. The theory is out of control
as argued by Banks, Peskin and Susskind, and ’t Hooft. Throughout this
period, a few theorists, including ’t Hooft and Susskind, have felt that the
basic principles of quantum mechanics and statistical mechanics have to be
made to co-exist with black hole evaporation.
’t Hooft has argued that by resolving the paradox and removing the
contradiction, the way to the new paradigm will be opened. The main
purpose of this book is to lay out this case.
A second purpose involves development of string theory as a unified description of elementary particles, including their gravitational interactions.
Although still very incomplete, string theory appears to be a far more consistent mathematical framework for quantum gravity than ordinary field
theory. It is therefore worth exploring the differences between string theory and field theory in the context of black hole paradoxes. Quite apart
from the question of the ultimate correctness and consistency of string theory, there are important lessons to be drawn from the differences between
these two theories. As we shall see, although string theory is usually well
approximated by local quantum field theory, in the neighborhood of a black
hole horizon the differences become extreme. The analysis of these differences suggests a resolution of the black hole dilemma and a completely new
view of the relations between space, time, matter, and information.
The quantum theory of black holes, with or without strings, is far from
being a textbook subject with well defined rules. To borrow words from
Sidney Coleman, it is a “trackless swamp” with many false but seductive
paths and no maps. To navigate it without disaster we will need some
beacons in the form of trusted principles that we can turn to for direction.
In this book the absolute truth of the following four propositions will be
x
Black Holes, Information, and the String Theory Revolution
assumed: 1) The formation and evaporation of a black hole is consistent
with the basic principles of quantum mechanics. In particular, this means
that observations performed by observers who remain outside the black
hole can be described by a unitary time evolution. The global process,
beginning with asymptotic infalling objects and ending with asymptotic
outgoing evaporation products is consistent with the existence of a unitary S-matrix. 2) The usual semiclassical description of quantum fields
in a slowly varying gravitational background is a good approximation to
certain coarse grained features of the black hole evolution. Those features
include the thermodynamic properties, luminosity, energy momentum flux,
and approximate black body character of Hawking radiation. 3) Thirdly we
assume the usual connection between thermodynamics and quantum statistical mechanics. Thermodynamics results from coarse graining a more
microscopic description so that states with similar macroscopic behavior
are lumped into a single thermodynamic state. The existence of a thermodynamics will be taken to mean that a microscopic set of degrees of freedom exists whose coarse graining leads to the thermal description. More
specifically we assume that a thermodynamic entropy S implies that approximately exp(S) quantum states have been lumped into one thermal
state.
These three propositions, taken by themselves, are in no way radical.
Proposition 1 and 3 apply to all known forms of matter. Proposition 2
may perhaps be less obvious, but it nevertheless rests on well-established
foundations. Once we admit that a black hole has energy, entropy, and
temperature, it must also have a luminosity. Furthermore the existence of a
thermal behavior in the vicinity of the horizon follows from the equivalence
principle as shown in the fundamental paper of Unruh. Why then should
any of these principles be considered controversial? The answer lies in a
fourth proposition which seems as inevitable as the first three: 4) The
fourth principle involves observers who fall through the horizon of a large
massive black hole, carrying their laboratories with them. If the horizon
scale is large enough so that tidal forces can be ignored, then a freely
falling observer should detect nothing out of the ordinary when passing the
horizon. The usual laws of nature with no abrupt external perturbations
will be found valid until the influence of the singularity is encountered. In
considering the validity of this fourth proposition it is important to keep in
mind that the horizon is a global concept. The existence, location, size, and
shape of a horizon depend not only on past occurrences, but also on future
events. We ourselves could right now be at the horizon of a gigantic black
Preface
xi
hole caused by matter yet to collapse in the future. The horizon in classical
relativity is simply the mathematical surface which separates those points
from which any light ray must hit a singularity from those where light may
escape to infinity. A mathematical surface of this sort should have no local
effect on matter in its vicinity.
In Chapter 9 we will encounter powerful arguments against the mutual
consistency of propositions 1–4. The true path through the swamp at times
becomes so narrow it seems to be a dead end, while all around false paths
beckon. Beware the will-o’-the-wisp and don’t lose your nerve.
Contents
Preface
vii
Part 1: Black Holes and Quantum Mechanics
1
1. The Schwarzschild Black Hole
3
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Schwarzschild Coordinates . . . . . . . . . . . .
Tortoise Coordinates . . . . . . . . . . . . . . .
Near Horizon Coordinates (Rindler space) . . .
Kruskal–Szekeres Coordinates . . . . . . . . . .
Penrose Diagrams . . . . . . . . . . . . . . . .
Formation of a Black Hole . . . . . . . . . . . .
Fidos and Frefos and the Equivalence Principle
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2. Scalar Wave Equation in a Schwarzschild Background
25
2.1 Near the Horizon . . . . . . . . . . . . . . . . . . . . . . . .
3. Quantum Fields in Rindler Space
3.1
3.2
3.3
3.4
3.5
Classical Fields . . . . . . . .
Entanglement . . . . . . . . .
Review of the Density Matrix
The Unruh Density Matrix .
Proper Temperature . . . . .
28
31
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4. Entropy of the Free Quantum Field in Rindler Space
4.1 Black Hole Evaporation . . . . . . . . . . . . . . . . . . . .
xiii
3
7
8
10
14
15
21
31
32
34
36
39
43
48
xiv
Black Holes, Information, and the String Theory Revolution
5. Thermodynamics of Black Holes
51
6. Charged Black Holes
55
7. The Stretched Horizon
61
8. The Laws of Nature
69
8.1
8.2
8.3
8.4
Information Conservation
Entanglement Entropy . .
Equivalence Principle . .
Quantum Xerox Principle
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9. The Puzzle of Information Conservation in Black Hole
Environments
9.1
9.2
9.3
69
71
77
79
81
A Brick Wall? . . . . . . . . . . . . . . . . . . . . . . . . .
Black Hole Complementarity . . . . . . . . . . . . . . . .
Baryon Number Violation . . . . . . . . . . . . . . . . . .
10. Horizons and the UV/IR Connection
84
85
89
95
Part 2: Entropy Bounds and Holography
99
11. Entropy Bounds
101
11.1
11.2
11.3
11.4
11.5
11.6
Maximum Entropy . . . . . . . . . . .
Entropy on Light-like Surfaces . . . . .
Friedman–Robertson–Walker Geometry
Bousso’s Generalization . . . . . . . . .
de Sitter Cosmology . . . . . . . . . . .
Anti de Sitter Space . . . . . . . . . . .
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12. The Holographic Principle and Anti de Sitter Space
12.1 The Holographic Principle . . . . .
12.2 AdS Space . . . . . . . . . . . . . .
12.3 Holography in AdS Space . . . . . .
12.4 The AdS/CFT Correspondence . .
12.5 The Infrared Ultraviolet Connection
12.6 Counting Degrees of Freedom . . .
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101
105
110
114
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123
127
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127
128
130
133
135
138
Contents
xv
13. Black Holes in a Box
13.1
13.2
141
The Horizon . . . . . . . . . . . . . . . . . . . . . . . . . 144
Information and the AdS Black Hole . . . . . . . . . . . 144
Part 3: Black Holes and Strings
149
14. Strings
14.1
14.2
14.3
14.4
Light Cone Quantum Mechanics
Light Cone String Theory . . .
Interactions . . . . . . . . . . .
Longitudinal Motion . . . . . .
151
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153
156
159
161
15. Entropy of Strings and Black Holes
165
Conclusions
175
Bibliography
179
Index
181
PART 1
Black Holes and Quantum Mechanics
Chapter 1
The Schwarzschild Black Hole
Before beginning the study of the quantum theory of black holes, one
must first become thoroughly familiar with the geometry of classical black
holes in a variety of different coordinate systems. Each coordinate system
that we will study has its own particular utility, and no one of them is in
any sense the best or most correct description. For example, the Kruskal–
Szekeres coordinate system is valuable for obtaining a global overview of the
entire geometry. It can however be misleading when applied to observations
made by distant observers who remain outside the horizon during the entire
history of the black hole. For these purposes, Schwarzschild coordinates, or
the related tortoise coordinates, which cover only the exterior of the horizon
are in many ways more valuable.
We begin with the simplest spherically symmetric static uncharged
black holes described by Schwarzschild geometry.
1.1
Schwarzschild Coordinates
In Schwarzschild coordinates, the Schwarzschild geometry is manifestly
spherically symmetric and static. The metric is given by
dτ 2 = (1 −
2MG
2
r )dt
− (1 −
2MG −1 2
dr
r )
− r2 dΩ2
(1.1.1)
= gµν dxµ dxν .
where dΩ2 ≡ dθ2 + sin2 θdφ2 .
The coordinate t is called Schwarzschild time, and it represents the time
recorded by a standard clock at rest at spatial infinity. The coordinate r
is called the Schwarzschild radial coordinate. It does not measure proper
3
4
Black Holes, Information, and the String Theory Revolution
spatial distance from the origin, but is defined so that the area of the 2sphere at r is 4πr2 . The angles θ, φ are the usual polar and azimuthal
angles. In equation 1.1.1 we have chosen units such that the speed of light
is 1.
The horizon, which we will tentatively define as the place where g00
vanishes, is given by the coordinate r = 2M G. At the horizon grr becomes
singular. The question of whether the geometry is truly singular at the
horizon or if it is the choice of coordinates which are pathological is subtle.
In what follows we will see that no local invariant properties of the geometry
are singular at r = 2M G. Thus a small laboratory in free fall at r =
2M G would record nothing unusual. Nevertheless there is a very important
sense in which the horizon is globally special if not singular. To a distant
observer the horizon represents the boundary of the world, or at least that
part which can influence his detectors.
To determine whether the local geometry is singular at r = 2M G we can
send an explorer in from far away to chart it. For simplicity let’s consider
a radially freely falling observer who is dropped from rest from the point
r = R. The trajectory of the observer in parametric form is given by
R
τ=
2
t = ( R2 + 2M G)
r=
R
(1 + cosη)
2
1/2
R
2MG
R
2M G
−1
(1.1.2)
(η + sinη)
1/2
η +
R
2
( R −1)1/2 + tan η2 +2M G log 2MRG
( 2M G −1)1/2 −tan η2 R
2MG
(1.1.3)
−1
1/2
sinη
(1.1.4)
[0 < η < π]
where τ is the proper time recorded by the observer’s clock. From these
overly complicated equations it is not too difficult to see that the observer
arrives at the point r = 0 after a finite interval
π
τ = R
2
R
2M G
12
(1.1.5)
Evidently the proper time when crossing the horizon is finite and smaller
than the expression in equation 1.1.5.
The Schwarzschild Black Hole
5
What does the observer encounter at the horizon? An observer in free
fall is not sensitive to the components of the metric, but rather senses the
tidal forces or curvature components. Define an orthonormal frame such
that the observer is momentarily at rest. We can construct unit basis
vectors, τ̂ , ρ̂, θ̂, φ̂ with τ̂ oriented along the observer’s instantaneous time
axis, and ρ̂ pointing radially out. The non-vanishing curvature components
are given by
Rτ̂ θ̂τ̂ θ̂ = Rτ̂ φ̂τ̂ φ̂ = −Rρ̂θ̂ρ̂θ̂ = −Rρ̂φ̂ρ̂φ̂ =
MG
r3
(1.1.6)
Rθ̂φ̂θ̂τ̂ = −Rτ̂ ρ̂τ̂ ρ̂ =
2MG
r3
Thus all the curvature components are finite and of order
R(Horizon) ∼
1
M 2 G2
(1.1.7)
at the horizon. For a large mass black hole they are typically very small.
Thus the infalling observer passes smoothly and safely through the horizon.
On the other hand the tidal forces diverge as r → 0 where a true local
singularity occurs. At this point the curvature increases to the point where
the classical laws of nature must fail.
Let us now consider the history of the infalling observer from the viewpoint of a distant observer. We may suppose that the infalling observer
sends out signals which are received by the distant observer. The first
surprising thing we learn from equations 1.1.2, 1.1.3, and 1.1.4 is that the
crossing of the horizon does not occur at any finite Schwarzschild time. It is
easily seen that as r tends to 2M G, t tends to infinity. Furthermore a signal
originating at the horizon cannot reach any point r > 2M G until an infinite Schwarzschild time has elapsed. This is shown in Figure 1.1. Assuming
that the infalling observer sends signals at a given frequency ν, the distant
observer sees those signals with a progressively decreasing frequency. Over
the entire span of Schwarzschild time the distant observer records only a
finite number of pulses from the infalling transmitter. Unless the infalling
observer increases the frequency of his/her signals to infinity as the horizon
is approached, the distant observer will inevitably run out of signals and
lose track of the transmitter after a finite number of pulses. The limits
imposed on the information that can be transmitted from near the horizon
are not so severe in classical physics as they are in quantum theory. According to classical physics the infalling observer can use an arbitrarily large
carrier frequency to send an arbitrarily large amount of information using
6
Black Holes, Information, and the String Theory Revolution
b
c = Signal
originating
near horizon
d
c
d = Signal
from infalling
to distant
Black
Hole
a = distant
observer
a
b = infalling
observer
Fig. 1.1
b
Infalling observer sending signals to distant Schwarzschild observer
an arbitrarily small energy without significantly disturbing the black hole
and its geometry. Therefore, in principle, the distant observer can obtain
information about the neighborhood of the horizon and the infalling system right up to the point of horizon crossing. However quantum mechanics
requires that to send even a single bit of information requires a quantum of
energy. As the observer approaches the horizon, this quantum must have
higher and higher frequency, implying that the observer must have had a
large energy available. This energy will back react on the geometry, dis-
The Schwarzschild Black Hole
7
turbing the very quantity to be measured. Thereafter, as we shall see, no
information can be transmitted from behind the horizon.
1.2
Tortoise Coordinates
A change of radial coordinate maps the horizon to minus infinity so that
the resulting coordinate system covers only the region r > 2M G. We define
the tortoise coordinate r∗ by
1
1−
2MG
r
2M G
dr = 1 −
(dr∗ )2
r
2
(1.2.8)
so that
dτ
2
2M G
= 1−
[dt2 − (dr∗ )2 ] − r2 dΩ2
r
(1.2.9)
The interesting point is that the radial-time part of the metric now has a
particularly simple form, called conformally flat . A space is called conformally flat if its metric can be brought to the form
dτ 2 = F (x) dxµ dxν ηµν
(1.2.10)
with ηµν being the usual Minkowski metric. Any two-dimensional space
is conformally flat, and a slice through Schwarzschild space at fixed θ, φ
is no exception. In equation 1.2.9 the metric of such a slice is manifestly
conformally flat. Furthermore it is also static.
The tortoise coordinate r∗ is given explicitly by
r∗ = r + 2M G log
r − 2M G
2M G
(1.2.11)
Note:
r∗ → −∞ at the horizon.
We shall see that wave equations in the black hole background have a very
simple form in tortoise coordinates.
8
1.3
Black Holes, Information, and the String Theory Revolution
Near Horizon Coordinates (Rindler space)
The region near the horizon can be explored by replacing r by a coordinate ρ which measures proper distance from the horizon:
r
ρ = 2MG grr (r ) dr
=
=
r
2MG
(1 −
2MG − 12
r )
dr
(1.3.12)
r
r (r − 2M G) + 2M G sinh−1 ( 2MG
− 1)
In terms of ρ and t the metric takes the form
2M G
2
dτ = 1 −
dt2 − dρ2 − r(ρ)2 dΩ2
r(ρ)
Near the horizon equation 1.3.12 behaves like
ρ ≈ 2 2M G(r − 2M G)
(1.3.13)
(1.3.14)
giving
dτ 2 ∼
= ρ2
dt
4M G
2
− dρ2 − r2 (ρ) dΩ2
(1.3.15)
Furthermore, if we are interested in a small angular region of the horizon
arbitrarily centered at θ = 0 we can replace the angular coordinates by
Cartesian coordinates
x = 2M G θ cosφ
(1.3.16)
y = 2M G θ sinφ
Finally, we can introduce a dimensionless time ω
ω =
t
4M G
(1.3.17)
and the metric then takes the form
dτ 2 = ρ2 dω 2 − dρ2 − dx2 − dy 2
(1.3.18)
It is now evident that ρ and ω are radial and hyperbolic angle variables
for an ordinary Minkowski space. Minkowski coordinates T , Z can be
The Schwarzschild Black Hole
9
defined by
T = ρ sinhω
(1.3.19)
Z = ρ coshω
to get the familiar Minkowski metric
dτ 2 = dT 2 − dZ 2 − dX 2 − dY 2
(1.3.20)
It should be kept in mind that equation 1.3.20 is only accurate near r =
2M G, and only for a small angular region. However it clearly demonstrates
that the horizon is locally nonsingular, and, for a large black hole, is locally
almost indistinguishable from flat space-time.
In Figure 1.2 the relation between Minkowski coordinates and the ρ, ω
coordinates is shown. The entire Minkowski space is divided into four
quadrants labeled I, II, III, and IV. Only one of those regions, namely
II
ω=ω 2
ω=ω 1
III
t=0
I
ρ=ρ1
IV
Fig. 1.2
ρ=ρ2
Relation between Minkowski and Rindler coordinates
10
Black Holes, Information, and the String Theory Revolution
Region I lies outside the black hole horizon. The horizon itself is the
origin T = Z = 0. Note that it is a two-dimensional surface in the fourdimensional space-time. This may appear surprising, since originally the
horizon was defined by the single constraint r = 2M G, and therefore appears to be a three dimensional surface. However, recall that at the horizon
g00 vanishes. Therefore the horizon has no extension or metrical size in the
time direction.
The approximation of the near-horizon region by Minkowski space is
called the Rindler approximation. In particular the portion of Minkowski
space approximating the exterior region of the black hole, i.e. Region I, is
called Rindler space. The time-like coordinate, ω, is called Rindler time.
Note that a translation of Rindler time ω → ω + constant is equivalent to
a Lorentz boost in Minkowski space.
1.4
Kruskal Szekeres Coordinates
Finally we can bring the black hole metric to the form
dτ 2 = F (R) [R2 dω 2 − dR2 ] − r2 dΩ2
(1.4.21)
For small ρ equation 1.3.15 shows that ρ ≈ R. A more accurate comparison
with the original Schwarzschild metric gives the following requirements:
2M G
R2 F (R) = 16M 2 G2 1 −
(1.4.22)
r
F (R) dR2 =
1
1−
from which it follows that
R
4M G log
= r + 2M G log
MG
or
R = M G exp
2MG
r
dr2
r − 2M G
2M G
r∗
4M G
(1.4.23)
= r∗
(1.4.24)
(1.4.25)
R and ω can be thought of as radial and hyperbolic angular coordinates
of a space which is conformal to flat 1+1 dimensional Minkowski space.
The Schwarzschild Black Hole
11
Letting
R eω = V
(1.4.26)
R e−ω = −U
be “radial light-like” variables, the radial-time part of the metric takes the
form
dτ 2 = F (R) dU dV
(1.4.27)
The coordinates U , V are shown in Figure 1.3. The surfaces of constant
r are the timelike hyperbolas in Figure 1.3. As r tends to 2M G the hyperbolas become the broken straight lines H + and H − which we will call
the extended past and future horizons. Although the extended horizons
lie at finite values of the Kruskal–Szekeres coordinates, they are located
at Schwarzschild time ±∞. Thus we see that a particle trajectory which
crosses H + in a finite proper time, crosses r = 2M G only after an infinite
Schwarzschild time.
U
V
+
H
II
III
I
r =
con
nt
sta
Fig. 1.3
H
IV
U, V Kruskal–Szekeris coordinates
The region of Schwarzschild space with r < 2M G can be taken
to be Region II. In this region the surfaces of constant r are the
12
Black Holes, Information, and the String Theory Revolution
spacelike hyperboloids
U V = positive constant
(1.4.28)
2
The true singularity at r = 0 occurs at R2 = − (M G) , or
2
U V = (M G)
(1.4.29)
R2
U
8
y
VV = -(MG 2
ularit
VV
eSing VVVV
VV ) = -UV
Futur
V
VVV
V
VVVVVVVVVVVV
r
t= =2M
G
The entire maximal analytic extension of the Schwarzschild geometry is
easily described in Kruskal–Szekeres coordinates. It is shown in Figure 1.4.
V
r =
con
stan
t
H
or
i
H +zon
II
I
External
III
Ho
riz
on
-H
IV
G
2M
r= t=
VVVVVVVVVVVVV
VVV
VV
V
V
VV
Past Singularity
VV
V
VV
8
Fig. 1.4
Maximal analytic extension of Schwarzschild in Kruskal–Szekeris coor-
dinates
A useful property of Kruskal–Szekeres coordinates is the fact that light
rays and timelike trajectories always lie within a two-dimensional light cone
bounded by 45o lines. A radial moving light ray travels on a trajectory
V = constant or U = constant. A nonradially directed light ray or timelike trajectory always lies inside the two-dimensional light cone. With this
in mind, it is easy to understand the causal properties of the black hole
The Schwarzschild Black Hole
13
II
P
2
V
O
ut
lig rad goin
ht ial g
ra
y
V
Future Singularity
VV
VV
V VV 2
VV
2
V
V
VRV=
-(MG)
=
-UV
VVVVVVVVVV
H
or
H + izo
n
U
8
VV
V
r=
2
t= MG
geometry. Consider a point P1 in Region I. A radially outgoing light ray
from P1 will escape falling into the singularity as shown in Figure 1.5. An
incoming light ray from P1 will eventually cross H + and then hit the future
singularity. Thus an observer in Region I can send messages to infinity as
well as into Region II.
ng
lli
fa ial ay
In rad t r
h
lig
P
1
III
H
or
i
H - zon
I
External
IV
r=
t= 2M
- G
8
VV
VVVVVVVV
VVVV
V VV
V VV
VV
V
VV
Past Singularity
VV
Fig. 1.5 Radial light rays from a point in Region I and Region II using Kruskal–
Szekeris coordinates
Consider next Region II. From any point P2 any signal must eventually
hit the singularity. Furthermore, no signal can ever escape to Region I.
Thus no observer who stays outside r = 2M G can ever be influenced by
events in Region II. For this reason Region II is said to be behind the
horizon. Regions III and IV, as we will see, are not relevant to the classical
problem of black holes formed by collapsing matter. Nevertheless let us
consider them. From Region III no signal can ever get to Region I, and so
it is also behind the horizon. On the other hand, points in Region IV can
14
Black Holes, Information, and the String Theory Revolution
communicate with Region I. Region I however cannot communicate with
Region IV. All of this is usually described by saying that Regions II and
III are behind the future horizon while Regions III and IV are behind the
past horizon.
1.5
Penrose Diagrams
Penrose diagrams are a useful way to represent the causal structure
of spacetimes, especially if, like the Schwarzschild black hole, they have
spherical symmetry. They represent the geometry of a two-dimensional
surface of fixed angular coordinates. Furthermore they “compactify” the
geometry so that it can be drawn in total on the finite plane. As an example,
consider ordinary flat Minkowski space. Ignoring angular coordinates,
dτ 2 = dt2 − dr2 − angular part = (dt + dr)(dt − dr) − angular part
(1.5.30)
Radial light rays propagate on the light cone dt ± dr = 0.
Any transformation that is of the form
Y + = F (t + r)
Y − = F (t − r)
(1.5.31)
will preserve the form of the light cone. We can use such a transformation
to map the entire infinite space 0 ≤ r ≤ ∞, −∞ ≤ t ≤ +∞ to a finite
portion of the plane. For example
Y + = tanh(t + r)
Y − = tanh(t − r)
(1.5.32)
The entire space-time is mapped to the finite triangle bounded by
Y+ = 1
Y − = −1
+
Y − Y− = 0
(1.5.33)
as shown in Figure 1.6. Also shown in Figure 1.6 are some representative
contours of constant r and t.
There are several infinities on the Penrose diagram. Future and past
time-like infinities (t = ±∞) are the beginnings and ends of time-like trajectories. Space-like infinity (r = ∞) is where all space-like surfaces end.
In addition to these there are two other infinities which are called I ± .
The Schwarzschild Black Hole
8
t=+
15
Y +=+1
+
t=3
t=2
t=1
r=
8
t=0
r=
3
r=1
r=2
r=0
Y +=Y -
Y - =-1
Fig. 1.6
8
t=-
Penrose diagram for Minkowski space
They are past and future light-like infinity, and they represent the origin
of incoming light rays and the end of outgoing light rays.
Similar deformations can be carried out for more interesting geometries,
such as the black hole geometry represent by Kruskal–Szekeres coordinates.
The resulting Penrose diagram is shown in Figure 1.7.
1.6
Formation of a Black Hole
The eternal black hole described by the static Schwarzschild geometry
is an idealization. In nature, black holes are formed from the collapse of
gravitating matter. The simpest model for black hole formation involves a
collapsing thin spherical shell of massless matter. For example, a shell of
photons, gravitons, or massless neutrinos with very small radial extension
and total energy M provides an example.
To construct the geometry, we begin with the empty space Penrose
diagram with the infalling shell represented by an incoming light-like line
16
Black Holes, Information, and the String Theory Revolution
t=+
8
Future Singularity
+
I
III
r =
8
H+
II
H
-
t=-
8
IV
Past Singularity
r
2
t=+
8
Future Singularity
II
H
or
iz
on
r =
r
r 1=
+
t=2
t=0
r=
t=-1
t=-2
III
8
t=1
on
iz
or
H
-
t=-
8
IV
Past Singularity
Penrose diagram for Schwarzschild black hole, showing regions (top) and
curves of fixed radial position and constant time (bottom)
Fig. 1.7
(see Figure 1.8). The particular value of Y + chosen for the trajectory is
arbitrary since any two such values are related by a time translation. The
infalling shell divides the Penrose diagram into two regions, A and B. The
Region A is interior to the shell and represents the initial flat space-time
The Schwarzschild Black Hole
17
YY+
B
A
-
Fig. 1.8 Minkowski space Penrose diagram for radially infalling spherical shell of
massless particles with energy M
before the shell passes. Region B is the region outside the shell and must
be modified in order to account for the gravitational field due to the mass
M.
In Newtonian physics the gravitational field exterior to a spherical mass
distribution is uniquely that of a point mass located at the center of the
distribution. Much the same is true in general relativity. In this case
Birkoff’s theorem tells us that the geometry outside the shell must be the
Schwarzschild geometry. Accordingly, we consider the Penrose diagram for
a black hole of mass M divided into regions A’ and B’ by an infalling
massless shell as in Figure 1.9. Once again the particular value of Y +
chosen for the trajectory is immaterial. Just as in Figure 1.8 where the
Region B is unphysical, in Figure 1.9 the Region A’ is to be discarded.
To form the full classical evolution the regions A of Figure 1.8 and B’ of
Figure 1.9 must be glued together. However this must be done so that
the “radius” of the local two sphere represented by the angular coordinates
(θ, φ) is continuous. In other words, the mathematical identification of
the boundaries of A and B’ must respect the continuity of the variable r.
Black Holes, Information, and the String Theory Revolution
Y
Y
+
-
18
B’
A’
-
Penrose diagram of Schwarzschild black hole with radially infalling shell
of massless particle with energy M
Fig. 1.9
Since in both cases r varies monotonically from r = ∞ at I − to r = 0,
the identification is always possible. One of the two Penrose diagrams will
have to undergo a deformation along the Y − direction in order to make
the identification smoothly, but this will not disturb the form of the light
cones. Thus in Figure 1.10 we show the resulting Penrose diagram for the
complete geometry. On Fig 1.10, a light-like surface H is shown as a dotted
line. It is clear that any light ray or timelike trajectory that originates to
the upper left of H must end at the singularity and cannot escape to I +
(or t = ∞). This identifies H as the horizon. In Region B’ the horizon
is identical to the surface H + of Figure 1.7, that is it coincides with the
future horizon of the final black hole geometry and is therefore found at
r = 2M G. On the other hand, the horizon also extends into the Region A
where the metric is just that of flat space-time. In this region the value of
r on the horizon grows from an initial value r = 0 to the value r = 2M G
at the shell.
It is evident from this discussion that the horizon is a global and not
a local concept. In the Region A no local quantity will distinguish the
presence of the horizon whose occurence is due entirely to the future collapse
of the shell.
Consider next a distant observer located on a trajectory with r >>
2M G. The observer originates at past time-like infinity and eventually
ends at future time-like infinity, as shown in Figure 1.11. The distant
observer collects information that arrives at any instant from his backward
light cone. Evidently such an observer never actually sees events on the
The Schwarzschild Black Hole
19
areas
Match
here
of 2-sp
B’
A
r
<
t=
G
M
H
r
2M
=2
8
r=0
+
G
r=0
-
Fig. 1.10
Penrose diagram for collapsing shell of massless particles
8
r=
Black Holes, Information, and the String Theory Revolution
Di
sta
nt
ob
se
rv
er
20
Fig. 1.11
Distant observer to collapsing spherical shell
horizon. In this sense the horizon must be regarded as at the end of time.
Any particle or wave which falls through the horizon is seen by the distant
observer as asymptotically approaching the horizon as it is infinitely red
shifted. At least that is the case classically.
This basic description of black hole formation is much more general than
might be guessed. It applies with very little modification to the collapse of
all kinds of massive matter as well as to non-spherical distributions. In all
cases the horizon is a lightlike surface which separates the space-time into
an inner and an outer region. Any light ray which originates in the inner
region can never reach future asymptotic infinity, or for that matter ever
reach any point of the outer region. The events in the outer region can
send light rays to I + and time-like trajectories to t = ∞.
The horizon, as we have seen, is a global concept whose location depends
on all future events. It is composed of a family of light rays or null geodesics,
passing through each space-time point on the horizon. This is shown in
Figure 1.12. Notice that null geodesics are vertical after the shell crosses
the horizon and essentially at 45o prior to that crossing. These light rays
are called the generators of the horizon.
Singularity forms
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
The Schwarzschild Black Hole
21
Horizon
Shell crosses
Horizon
Shell of
radially
moving
light-rays
Horizon Forms
Fig. 1.12
1.7
The horizon as family of null geodesics
Fidos and Frefos and the Equivalence Principle
In considering the description of events near the horizon of a static
black hole from the viewpoint of an external observer[1], it is useful to
imagine space to be filled with static observers, each located at a fixed
(r, θ, φ). Such observers are called fiducial observers, or by the whimsical
abbreviation, FIDOS. Each Fido carries a clock which may be adjusted to
record Schwarzschild time t. This means that Fidos at different r values
see their own clocks running at different proper rates. Alternatively, they
could carry standard clocks which always record proper time τ . At a given
r the relation between Schwarzschild time t and the Fidos proper time τ is
given by
√
2M G 12
dτ
= g00 = [1 −
]
dt
r
(1.7.34)
Thus, to the Fido near r = 2M G, the Schwarzschild clock appears to run
at a very rapid rate. Another possible choice of clocks would record the
dimensionless hyperbolic angle ω defined by equation 1.3.17.
The spatial location of the Fidos can be labeled by the angular coordinates (θ, φ) and any one of the radial variables r, r∗ , or ρ. Classically the
Fidos can be thought of as mathematical fictions or real but arbitrarily light
22
Black Holes, Information, and the String Theory Revolution
systems suspended by arbitrarily light threads from some sort of suspension
system built around the black hole at a great distance. The acceleration
of a Fido at proper distance ρ is given by ρ1 for ρ << M G. Quantum mechanically we have a dilemma if we try to imagine the Fidos as real. If they
are extrememly light their locations will necessarily suffer large quantum
fluctuations, and they will not be useful as fixed anchors labeling spacetime points. If they are massive they will influence the gravitational field
that we wish to describe. Quantum mechanically, physical Fidos must be
replaced by a more abstract concept called gauge fixing. The concept of
gauge fixing in gravitation theory implies a mathematical restriction on the
choice of coordinates. However all real observables are required to be gauge
invariant.
Now let us consider a classical particle falling radially into a black hole.
There are two viewpoints we can adopt toward the description of the particle’s motion. The first is the viewpoint of the Fidos who are permanently
stationed outside the black hole. It is a viewpoint which is also useful to
a distant observer, since any observation performed by a Fido can be communicated to distant observers. According to this viewpoint, the particle
never crosses the horizon but asymptotically approaches it. The second
viewpoint involves freely falling observers (FREFOS) who follow the particle as it falls. According to the Frefos, they and the particle cross the
horizon after a finite time. However, once the horizon is crossed, their
observations cannot be communicated to any Fido or to a distant observer.
Once the infalling particle is near the horizon its motion can be described
by the coordinates (T, Z, X, Y ) defined in equations 1.3.16 and 1.3.19. Since
the particle is freely falling, in the Minkowski coordinates its motion is a
straight line
dZ
dτ
=
dT
dτ
pZ
m
=
= − pmZ
pT
m
(1.7.35)
where pZ and pT are the Z and T components of momentum, and m is
the mass of the particle. As the particle freely falls past the horizon, the
components pZ and pT may be regarded as constant or slowly varying.
They are the components seen by Frefos.
The components of momentum seen by Fidos are the components pρ
and pτ which, using equation 1.3.19, are given by
pρ = pZ coshω + pT sinhω
pτ = pZ sinhω + pT coshω
(1.7.36)
The Schwarzschild Black Hole
23
For large times we find
t
)
(1.7.37)
4M G
Thus we find the momentum of an infalling particle as seen by a Fido
grows exponentially with time! It is also easily seen that ρ, the proper
spatial distance of the particle from the horizon, exponentially decreases
with time
t
)
(1.7.38)
ρ(t) ≈ ρ(0)exp(−
4M G
Locally the relation between the coordinates of the Frefos and Fidos is
a time dependent boost along the radial direction. The hyperbolic boost
angle is the dimensionless time ω. Eventually, during the lifetime of the
black hole this boost becomes so large that the momentum of an infalling
particle (as seen by a Fido) quickly exceeds the entire mass of the universe.
As a consequence of the boost, the Fidos see all matter undergoing
Lorentz contraction into a system of arbitrarily thin “pancakes” as it approaches the horizon. According to classical physics, the infalling matter is
stored in “sedimentary” layers of diminishing thickness as it eternally sinks
toward the horizon (see Figure 1.13). Quantum mechanically we must expect this picture to break down by the time the infalling particle has been
squeezed to within a Planck distance from the horizon. The Frefos of course
see the matter behaving in a totally unexceptional way.
Horizo
n
pρ ≈ pτ ≈ 2pZ expω = 2pZ exp(
Fig. 1.13
Sedimentary layers of infalling matter on horizon
24
Black Holes, Information, and the String Theory Revolution
Chapter 2
Scalar Wave Equation in a
Schwarzschild Background
In Chapters 3 and 4 we will be concerned with the behavior of quantum
fields near horizons. In this lecture we will study the properties of a scalar
wave equation in the background of a black hole.
Let us consider a conventional massless free Klein–Gordon field χ in the
Schwarzschild background. Here we will find great advantage in utilizing
tortoise coordinates in which the metric has the form
dτ 2 = F (r∗ ) [dt2 − (dr∗ )2 ] − r2 [dθ2 + sin2 θdφ2 ]
(2.0.1)
The action for χ is
1
2
I =
=
1
2
√
−g g µν ∂µ χ ∂ν χ d4 x
dt dr∗ dθ dφ { (∂t χ)
2
− r12 ( ∂χ
∂θ ) −
1
r 2 sin2 θ
2
− (∂r∗ χ)2
F
(2.0.2)
2
2
( ∂χ
∂φ ) } F r sinθ
Now define
ψ = rχ
(2.0.3)
and the action takes the form
I =
−
F
r2
(sinθ
1
2
∂ψ
[(∂t ψ)2 − ( ∂r
∗ −
∂ψ
∂θ
2
+
1
sinθ
25
∂ψ
∂φ
∂(lnr)
2
∂r ∗ ψ)
2
(2.0.4)
∗
)] dt dr dθ dφ
26
Black Holes, Information, and the String Theory Revolution
which, after an integration by parts and the introduction of spherical harmonic decomposition becomes
1
m 2
I =
−{
∂lnr 2
∂r ∗
+
∂
∂r ∗
[(ψ̇
2
m)
−
∂lnr 2
∂r ∗ } ψ m −
∂ψm
∂r ∗
2
+
(2.0.5)
F
r2
( +
1) ψ 2m ]dt dr∗
Using the relation between r and r∗
r∗ = r + 2M G ln(r − 2M G)
gives for each , m an action
2
2
∂ψ m
∂ψ m
1
∗
∗
2
Im =
−
− V (r ) ψ m
dt dr
2
∂t
∂r∗
where the potential V (r∗ ) is given by
( + 1)
r − 2M G
2M G
V (r∗ ) =
+
r
r2
r3
(2.0.6)
(2.0.7)
The equation of motion is
ψ̈
m
=
∂2ψ m
− V (r∗ ) ψ
(∂r∗ )2
(2.0.8)
m
and for a mode of frequency ν
−
∂ 2ψ m
+ V (r∗ ) ψ
(∂r∗ )2
m
= ν2 ψ
m
(2.0.9)
The potential V is shown in Figure 2.1 as a function of the Schwarzschild
coordinate r. For r >> 3M G the potential is repulsive. In fact it is
just the relativistic generalization of the usual repulsive centrifugal barrier.
However as the horizon is approached, gravitational attraction wins and the
potential becomes attractive, and pulls a wave packet toward the horizon.
The maximum of the potential, where the direction of the force changes,
depends weakly on the angular momentum . It is given by
1
14 2 + 14 + 9
1
−
1+ 1+
(2.0.10)
rmax = 3M G
2
9 2 ( + 1)2
2 ( + 1)
→ ∞ the maximum occurs at rmax ( → ∞) = 3M G.
The same potential governs the motion of massless classical particles.
One can see that the points rmax ( ) represent unstable circular orbits, and
For
Scalar Wave Equation in a Schwarzschild Background
27
V (r)
=2
Region of
Thermal
Atmosphere
=1
=0
r
2MG
Fig. 2.1
3MG
4MG
5MG
6MG
Effective potential for free scalar field vs Schwarzschild radial coordinate
the innermost such orbit is at r = 3M G. Any particle that starts with
vanishing radial velocity in the region r < 3M G will spiral into the horizon.
In the region of large negative r∗ where we approach the horizon, the
potential is unimportant, and the field behaves like a free massless Klein–
Gordon field. The eigenmodes in this region have the form of plane waves
which propagate with unit velocity
dr ∗
dt
= ∓1
(2.0.11)
ψ → ei k (r
∗
±t)
Let us consider a field quantum of frequency ν and angular momentum
propagating from large negative r∗ toward the barrier at r ≈ 3M G. Will it
pass over the barrier? To answer this we note that equation 2.0.9 has the
form of a Schrodinger equation for a particle of energy ν 2 in a potential V .
The particle has enough energy to overcome the barrier without tunneling
if ν 2 is larger than the maximum height of the barrier. For example, if
= 0 the height of the barrier is
Vmax
1
=
2M 2 G2
3
3
8
(2.0.12)
An s-wave quantum will therefore escape if
ν >
0.15
MG
(2.0.13)
28
Black Holes, Information, and the String Theory Revolution
0.15
Similarly an s-wave quantum with ν > MG
will be able to penetrate the
barrier from the outside and fall to the horizon. Less energetic particles
must tunnel through the barrier.
A particle of high angular momentum, whether on the inside or outside
of the barrier will have more difficulty penetrating through. For large
Vmax ≈
2
1
27 M 2 G2
(2.0.14)
Therefore the threshold energy for passing over the barrier is
1
ν ∼ √
27 M G
2.1
(2.0.15)
Near the Horizon
Near the horizon the exterior of the black hole may be described by the
Rindler metric
dτ = ρ2 dω 2 − dρ2 − dX 2 − dY 2
It is useful to replace ρ by a tortoise-like coordinate which again goes to
−∞ at the horizon. We define
u = logρ
and the metric near the horizon becomes
dτ 2 = exp(2u) dω 2 − du2 − dX 2 − dY 2
The scalar field action becomes
2
2
∂χ
∂χ
1
2
2u
dX dY du dω
I =
−
− e (∂⊥ χ)
2
∂ω
∂u
(2.1.16)
(2.1.17)
(2.1.18)
where ∂⊥ χ = (∂X , ∂Y ). Instead of using spherical waves, near the horizon
we can decompose χ into transverse plane waves with transverse wave vector
k⊥
χ =
d2 k⊥ ei k⊥ x⊥ χ(k⊥ , u, ω)
(2.1.19)
Scalar Wave Equation in a Schwarzschild Background
the action for a given wave number k is
1
I =
dω du (∂ω χ)2 − (∂u χ)2 − k 2 e2u χ2
2
29
(2.1.20)
Thus the potential is
V (k, u) = k 2 e2u
(2.1.21)
The correspondence between the momentum vector k and the angular
momentum is given by the usual connection between momentum and
angular momentum. If the horizon has circumference 2π(2M G), then
a wave with wave vector k⊥ will correspond to an angular momentum
| | = |k| r = 2M G|k|. Thus the potential in equation 2.1.21 is seen to
be proportional to 2 . For very low angular momentum the approximation
is not accurate, but qualitatively is correct for > 0. In approximating a
sum over and m by an integral over k, the integral should be infrared cut
1
.
off at |k| ∼ MG
From the action in equation 2.1.20 we obtain the equation of motion
∂2χ
∂2χ
−
+ k 2 exp(2u) χ = 0
∂ω 2
∂u2
(2.1.22)
A solution which behaves like exp(iνt) in Schwarzschild time has the
form
ei ν [4MGω] = ei λ ω
(2.1.23)
The time independent form of the equation of motion is
2
∂ 2χ
+
k
exp(2u)
χ = λ2 χ
(2.1.24)
∂u2
Once again we see that unless k = 0, there is a potential confining quanta to
the region near the horizon. Qualitatively, the behavior of a quantum field
in a black hole background differs from the Rindler space approximation
in that for the black hole, the potential barrier is cut off when ρ = eu
is greater than M G. By contrast, in the Rindler case V increases as eu
without bound.
We have not thus far paid attention to the boundary conditions at the
horizon where u → −∞. Since in this region the field χ(u) behaves like a
free massless field, the boundary condition would be expected to be that
the field is in the usual quantum ground state. In the next section we will
see that this is not so.
−
30
Black Holes, Information, and the String Theory Revolution
Chapter 3
Quantum Fields in Rindler Space
According to Einstein, the study of a phenomenon in a gravitational
field is best preceeded by a study of the same phenomenon in an accelerated
coordinate system. In that way we can use the special relativistic laws of
nature to understand the effect of a gravitational field.
As we have seen, the relativistic analogue of a uniformly accelerated
frame is Rindler space. Because Rindler space covers only a portion of the
space-time geometry (Region I) there are new and subtle features to the
description of quantum fields. These features are closely associated with
the existence of the horizon. The method we will use applies to any relativistic quantum field theory including those with nontrivial interactions.
For illustrative purposes we will consider a free scalar field theory. It is
important to bare in mind that such a non-interacting description is of limited validity. As we shall see, interactions become very important near the
horizon of a black hole. Ignoring them leads to an inconsistent description
of the Hawking evaporation process.
3.1
Classical Fields
First let us consider the evolution of a classical field in Rindler space.
The field in Region I of Figure 1.2 can be described in a self contained way.
Obviously influences from Regions II and III can never be felt in Region I
since no point in Regions II or III is in the causal past of any point in
Region I. Signals from Region IV can, of course, reach Region I, but to do
so they must pass through the surface ω = −∞. Therefore signals from
Region IV are regarded as initial data in the remote past by the Rindler
observer. Evidently the Rindler observer sees a world in which physical
phenomena can be described in a completely self contained way.
31
32
Black Holes, Information, and the String Theory Revolution
The evolution from one surface of constant ω to another is governed
by the Rindler Hamiltonian. Using conventional methods the generator of
ω-translations is given by
∞
HR =
dρ dX dY ρ T 00 (ρ, X, Y )
(3.1.1)
ρ=0
where T 00 is the usual Hamiltonian density used by the Minkowski observer.
For example, for a massive scalar field with potential V , T 00 is given by
T 00 =
1
Π2
2
+ (∇χ) + V (χ)
2
2
(3.1.2)
where Π is the canonical momentum conjugate to χ. The Rindler Hamiltonian is
2
2
∂χ
∂χ
ρ
2
Π +
HR =
dρ dx⊥
+
+ 2 V (χ)
(3.1.3)
2
∂ρ
∂x⊥
The origin of the factor ρ in the Rindler Hamiltonian density is straightforward. In Figure 3.1 the relation between neighboring equal Rindler-time
surfaces is shown. The proper time separation between the surfaces is
δτ = ρ δω
(3.1.4)
Thus, to push the ω-surface ahead requires a ρ-dependent time translation.
This is the reason that T 00 is weighted with the factor ρ. The Rindler
Hamiltonian is similar to the generator of Lorentz boosts from the viewpoint
of the Minkowski observer. However it only involves the degrees of freedom
in Region I.
3.2
Entanglement
Quantum fields can also be described in a self-contained fashion in
Rindler space, but a new twist is encountered. Our goal is to describe
the usual physics of a quantum field in Minkowski space, but from the
viewpoint of the Fidos in Region I, i.e. in Rindler space. To understand
the new feature, recall that in the usual vacuum state, the correlation between fields at different spatial points does not vanish. For example, in free
massless scalar theory the equal time correlator is given by
0| χ(X, Y, Z) χ(X , Y , Z ) |0 ∼
1
∆2
(3.2.5)
Quantum Fields in Rindler Space
33
,
t=
8
ω=
8
T
r > 2MG
p=
2
r < 2MG
ω=+4
p=
1
ω=+3
ω=+2
ω=+1
ω=0
Z
ρ=0
ω=−1
(Horizon)
ω=−2
Light
cone
ω=−3
ω=−4
Fig. 3.1
Equal time and proper distance surfaces in Rindler space
where ∆ is the space-like separation between the points (X, Y, Z) and
(X , Y , Z )
∆2 = (X − X )2 + (Y − Y )2 + (Z − Z )2
(3.2.6)
The two points might both lie within Region I, in which case the correlator in equation 3.2.5 represents the quantum correlation seen by Fido’s in
Region I. On the other hand, the two points might lie on opposite sides
of the horizon at Z = 0. In that case the correlation is unmeasurable to
the Fidos in Region I. Nevertheless it has significance. When two subsystems (fields in Regions I and III) become correlated, we say that they
are quantum entangled, so that neither can be described in terms of pure
states. The appropriate description of an entangled subsystem is in terms
of a density matrix.
34
3.3
Black Holes, Information, and the String Theory Revolution
Review of the Density Matrix
Suppose a system consists of two subsystems, A and B, which have
previously been in contact but are no longer interacting. The combined
system has a wavefunction
Ψ = Ψ(α, β)
(3.3.7)
where α and β are appropriate commuting variables for the subsystems A
and B.
Now suppose we are only interested in subsystem A. A complete description of all measurements of A is provided by the density matrix ρA (α, α ).
ρA (α, α ) =
Ψ∗ (α, β) Ψ(α , β)
(3.3.8)
β
Similarly, experiments performed on B are described by ρB (β, β ).
ρB (β, β ) =
Ψ∗ (α, β) Ψ(α, β )
(3.3.9)
α
The rule for computing an expectation value of an operator a composed
of A degrees of freedom is
a = T r a ρA
(3.3.10)
Density matrices have the following properties:
1)
T r ρ = 1 (total probability=1)
2)
ρ = ρ† (hermiticity)
3)
ρj ≥ 0 (all eigenvalues are positive or zero)
In the representation in which ρ is diagonal
⎛
⎞
ρ1 0 0 ... ...
⎜ 0 ρ2 0 ... ... ⎟
⎟
ρ = ⎜
⎝ 0 0 ρ3 ... ... ⎠
... ... ... ... ...
(3.3.11)
The eigenvalues ρj can be considered to be probabilities that the system
is in the j th state. However, unlike the case of a coherent superposition of
states, the relative phases between the states |j are random.
There is one special case when the density matrix is indistinguishable
from a pure state. This is the case in which only one eigenvalue ρj is
Quantum Fields in Rindler Space
35
nonzero. This case can only result from an uncorrelated product wave
function of the form
Ψ(α, β) = ψA (α) ψB (β)
(3.3.12)
A quantitative measure of the departure from a pure state is provided
by the Von Neumann entropy
S = −T r ρ log ρ = −
j
ρj log ρj .
(3.3.13)
S is zero if and only if all the eigenvalues but one are zero. The one nonvanishing eigenvalue is equal to 1 by virtue of the trace condition on ρ. The
entropy is also a measure of the degree of entanglement between A and B.
It is therefore called the entropy of entanglement.
The opposite extreme to a pure state is a completely incoherent density matrix in which all the eigenvalues are equal to N1 , where N is the
dimensionality of the Hilbert space. In that case S takes its maximum
value
1
1
log
= log N
(3.3.14)
Smax = −
N
N
j
More generally, if ρ is a projection operator onto a subspace of dimension
n, we find
S = log n
(3.3.15)
Thus we see that the Von Neumann entropy is a measure of the number
of states which have an appreciable probability in the statistical ensemble.
We may think of eS as an effective dimensionality of the subspace described
by ρ.
The Von Neumann (or entanglement) entropy should not be confused
with the thermal entropy of the second law of thermodynamics. This entropy has its origin in coarse graining. If a system with Hamiltonian H is
in thermal equilibrium at temperature T = 1/β then it is described by a
Maxwell–Boltzman density matrix
ρM.B. =
e−β H
.
T re−β H
(3.3.16)
In this case the thermal entropy is given by
Sthermal = −T r ρM.B. logρM.B.
(3.3.17)
36
3.4
Black Holes, Information, and the String Theory Revolution
The Unruh Density Matrix
Now let us consider the space of states describing a Lorentz invariant
quantum field theory in Minkowski space. In Figure 3.2 the surface T = 0
of Minkowski space is shown divided into two halves, one in Region I and
one in Region III. For the case of a scalar field χ the fields at each point
III
Fig. 3.2
χL
χR
I
Fields on the spacelike surface T = 0 in Minkowski space
of space form a complete set of commuting operators. These fields may be
decomposed into two subsets associated with regions I and III. we call them
χR and χL respectively. Thus
χ(X, Y, Z) = χR (X, Y, Z) Z > 0
(3.4.18)
χ(X, Y, Z) = χL (X, Y, Z) Z < 0
The general wave functional of the system is a functional of χL and χR
Ψ = Ψ(χL , χR )
(3.4.19)
We wish to compute the density matrix used by the Fidos in Region I to
describe their Rindler world. In particular we would like to understand the
density matrix ρR which represents the usual Minkowski vacuum to the
Fidos in Region I.
First let us see what we can learn from general principles. Obviously the
state Ψ is translationally invariant under the usual Minkowski space translations. Thus the Fidos must see the vacuum as invariant under translations
along the X and Y axes. However, the translation invariance along the Z
axis is explicitly broken by the act of singling out the origin Z = 0 for
Quantum Fields in Rindler Space
37
special consideration. From the X, Y translation invariance we conclude
that ρR commutes with the components of momentum in these directions
[pX , ρR ] = [pY , ρR ] = 0
(3.4.20)
A very important property of ρR is that it is invariant under Rindler
time translations ω → ω + constant. This follows from the Lorentz boost
invariance of Ψ. Thus
[HR , ρR ] = 0
(3.4.21)
To proceed further we must use the fact that Ψ(χL , χR ) is the ground
state of the Minkowski Hamiltonian. General path integral methods may
be brought to bare on the computation of the ground state wave functional.
Let us assume that the field theory is described in terms of an action
I =
d3 X dT L
(3.4.22)
The so-called Euclidean field theory is defined by replacing the time coordinate T by i X 0 . For example, the Euclidean version of ordinary scalar
field theory is obtained from the usual Minkowski action
I =
d3 X dT
1 2
χ̇ − (∇χ)2 − 2 V (χ)
2
(3.4.23)
Letting T → i X 0 we obtain the Euclidean action
IE =
d4 X
1 (∂X χ)2 − 2 V (χ)
2
(3.4.24)
Now a standard method of computing the ground state by path integration is to use the Feynman–Hellman theorem. Suppose we wish to compute
Ψ(χL , χR ). Then we consider the path integral
1
Ψ(χL , χR ) = √
Z
dχ(x) e−IE
(3.4.25)
where the path integral is over all χ(x) with X 0 > 0 and Z is an appropriate
normalization factor . The field χ(x) is constrained to equal (χL , χR ) on
the surface X 0 = 0. Finally the action IE is evaluated as an integral over
the portion of Minkowski space with X 0 > 0.
The boost invariance of the original Minkowski action insures that the
Euclidean action has four dimensional rotation invariance. In particular,
the invariance under ω-translations becomes invariance under rotations in
the Euclidean (Z, X 0 ) plane. This suggests a new way to carry out the
38
Black Holes, Information, and the String Theory Revolution
path integral. Let us define the Euclidean angle in the (Z, X 0 ) plane to be
θ. The angle θ is the Euclidean analogue of the Rindler time ω. Now let
us divide the region X 0 > 0 into infinitesimal angular wedges as shown in
Figure 3.3.
T
δθ
χL
Fig. 3.3
Horizon
ρ=0
χR
θ=0
Z
Euclidean analogue of Rindler space for path integration
The strategy for computing the path integral is to integrate over the
fields in the first wedge between θ = 0 and θ = δθ. The process can be
iterated until the entire region X 0 > 0 has been covered.
The integral over the first wedge is defined by constaining the fields at
θ = 0 and θ = δθ. This defines a transfer matrix G in the Hilbert space of
the field configuration χR . The matrix is recognized to be
G = (1 − δθ HR ) .
(3.4.26)
To compute the full path integral we raise the matrix G to the power
giving
1
Ψ(χL , χR ) = √ χL | e−π HR |χR
Z
π
δω
(3.4.27)
In other words, the path integral defining Ψ is computed as a transition matrix element between initial state χR and final state χL . The infinitesimal
generator which pushes θ surfaces forward is just the Rindler Hamiltonian.
Now we are prepared to compute the density matrix ρR . According to
Quantum Fields in Rindler Space
39
the definition in equation 3.3.8, the density matrix ρR is given by
ρR (χR , χR ) =
Ψ∗ (χL , χR ) Ψ(χL , χR ) dχL
(3.4.28)
Now using equation 3.4.27 we get
χR | e−π HR |χL χL | e−π HR |χR dχL
ρR (χR , χR ) = Z1
(3.4.29)
=
1
Z
χR | e
−2 π HR
|χR
In other words, the density matrix is given by the operator
ρR =
1
exp(−2 π HR )
Z
(3.4.30)
This remarkable result, discovered by William Unruh in 1976 , says that
the Fidos see the vacuum as a thermal ensemble with a density matrix of
the Maxwell–Boltzmann type. The temperature of the ensemble is
TR =
1
1
=
2π
βR
(3.4.31)
The derivation of the thermal character of the density matrix and the
value of the Rindler temperature in equation 3.4.31 is entirely independent
of the particulars of the relativistic field theory. It is equally correct for a
free scalar quantum field, quantum electrodynamics, or quantum chromodynamics.
3.5
Proper Temperature
It is noteworthy that the temperature TR is dimensionless. Ordinarily,
temperature has units of energy, or equivalently, inverse length. The origin
of the dimensionless temperature lies in the dimensionless character of the
Rindler time variable ω. Nevertheless we should be able to assign to each
Fido a conventional temperature that would be recorded by a standard
thermometer held at rest at the location of that Fido. We can consider
a thermometer to be a localized object with a set of proper energy levels
i . The levels i are the ordinary energy levels of the thermometer when it
is at rest. The thermometer is assumed to be very weakly coupled to the
quantum fields so that it eventually will come to thermal equilibrium with
them. Let us suppose that the thermometer is at rest with respect to the
Fido at position ρ so that it has proper acceleration ρ1 . The Rindler energy
40
Black Holes, Information, and the String Theory Revolution
in equation 3.1.1 evidently receives a contribution from the thermometer of
the form
ρ |i i| i
(3.5.32)
HR (thermometer) =
i
In other words the Rindler energy level of the ith state of the thermometer
is ρ i .
1
equilibrates with
When the quantum field at Rindler temperature 2π
the thermometer, the probability to find the thermometer excited to the
ith level is given by the Boltzmann factor
Pi =
e−2π ρ i
−2π ρ j
je
(3.5.33)
Accordingly, the thermometer registers a proper temperature
T (ρ) =
1
1
= TR
2π ρ
ρ
(3.5.34)
Thus each Fido experiences a thermal environment characterized by
a temperature which increases as we move toward the horizon at ρ = 0.
The proper temperature T (ρ) can also be expressed in terms of the proper
acceleration of the Fido which is equal to ρ1 . Thus, calling the acceleration
a, we find
T (ρ) =
a(ρ)
2π
(3.5.35)
The reader may wonder about the origin of the thermal fluctuations felt
by the Fidos, since the system under investigation is the Minkowski space
vacuum. The thermal fluctuations are nothing but the conventional virtual
vacuum fluctuations, but now being experienced by accelerated apparatuses. It is helpful in visualizing these effects to describe virtual vacuum
fluctuations as short lived particle pairs. In Figure 3.4 ordinary vacuum
fluctuations are shown superimposed on a Rindler coordinate mesh. One
virtual loop (a) is contained entirely in Region I. That fluctuation can be
thought of as a conventional fluctuation described by the quantum Hamiltonian HR . The fluctuation (b) contained in Region III has no significance
to the Fidos in Region I. Finally there are loops like (c) which are partly
in Region I but which also enter into Region III. These are the fluctuations
which lead to nontrivial entanglements between the degrees of freedom χL
and χR , and which cause the density matrix of Region I to be a mixed
state.
Quantum Fields in Rindler Space
41
8
ω=
(a)
(c)
(b)
8
ω=−
Fig. 3.4
Vacuum pair fluctuations near the horizon
A virtual fluctuation is usually considered to be short lived because it
“violates energy conservation”. If the virtual fluctuation of energy needed
to produce the pair is E, then the lifetime of the fluctuation ∼ E −1 .
Now consider the portion of the loop (b) which is found in Region I.
From the viewpoint of the Fidos, a particle is injected into the system at
ω = −∞ and ρ = 0. The particle travels to some distance and then falls
back towards ρ = 0 and ω = +∞. Thus, according to the Fidos, the
fluctuation lasts for an infinite time and is therefore not virtual at all. Real
particles are seen being injected into the Rindler space from the horizon,
and eventually fall back to the horizon. To state it differently, the horizon
behaves like a hot membrane radiating and reabsorbing thermal energy.
A natural question to ask is whether the thermal effects are “real”.
For example, we may ask whether any such thermal effects are seen by
freely falling observers carrying their thermometers with them as they pass
through the horizon. Obviously the answer is no. A thermometer at rest
in an inertial frame in the Minkowski vacuum will record zero temperature.
It is tempting to declare that the thermal effects seen by the Fidos are
fictitious and that the reality is best described in the frame of the Frefos.
42
Black Holes, Information, and the String Theory Revolution
However, by yielding to this temptation we risk prejudicing ourselves too
much toward the viewpoint of the Frefos. In particular, we are going to
encounter questions of the utmost subtlety concerning the proper relation
between events as seen by observers who fall through the horizon of a black
hole and those seen by observers who view the formation and evaporation
process from a distance. Thus for the moment, it is best to avoid the
metaphysical question of whose description is closer to reality. Instead we
simply observe that the phenomena are described differently in two different coordinate systems and that different physical effects are experienced
by Frefos and Fidos. In particular a Fido equipped with a standard thermometer, particle detector, or other apparatus, will discover all the physical
phenomena associated with a local proper temperature T (ρ) = 2π1 ρ . By
contrast, a Frefo carrying similar apparatuses will see only the zero temperature vacuum state. Later we will discuss the very interesting question
of how contradiction is avoided if a Frefo attempts to communicate to the
stationary Fidos the information that no thermal effects are present.
We can now state the sense in which a self contained description of
Rindler space is possible in ordinary quantum field theory. Since Rindler
space has a boundary at ρ = 0, a boundary condition of some sort must
be provided. We see that the correct condition must be that at some
small distance ρo , an effective “membrane” is kept at a fixed temperature
T (ρo ) = 2π1ρo by an infinite heat reservoir. It will prove useful later to
locate the membrane at a distance of order the Planck length P = G/c3
where quantum gravitational or string effects become important. Such a
fictitious membrane at Planckian distance from the horizon is called the
stretched horizon. We will see later that the stretched horizon has many
other physical properties besides temperature, although it is completely
unseen by observers who fall freely through it.
Chapter 4
Entropy of the Free Quantum Field in
Rindler Space
In the real world, a wide variety of different phenomena take place at
different temperature scales. At the lowest temperatures where only massless quanta are produced by thermal fluctuations, one expects to find a very
weakly interacting gas of gravitons, photons, and neutrinos. Increase the
temperature to the e+ , e− threshold and electron-positron pairs are produced. The free gas is replaced by a plasma. At higher temperatures, pions
are produced which eventually dissociate into quarks and gluons, and so it
goes, up the scale of energies. Finally, the Planck temperature is reached
where totally new phenomena of an as yet unimagined kind take place.
All of these phenomena have their place in the Fido’s description of the
region near a horizon. In this lecture we will consider an enormously oversimplified description of the world in which only a single free field is present
in a fixed space-time background. There is serious danger in extrapolating
far reaching conclusions from so oversimplified a situation. In fact, the paradoxes and contradictions associated with black holes, quantum mechanics,
and statistical thermodynamics that these lectures are concerned with are
largely a consequence of such unjustified extrapolation. Nevertheless, the
study of a free quantum field in Rindler space is a useful starting point.
We consider the field theory defined by equation 2.1.18. Fourier decomposing the field χ in equation 2.1.19 leads to the wave equation in equation
2.1.24.
−
2 2u ∂ 2 χk
+
k e
χk = λ2 χk
∂u2
(4.0.1)
In order to quantize the field χ it is necessary to provide a boundary condition when u → −∞. The simplest method of dealing with this region is to
introduce a cutoff at some point uo = log at which point the field (or its
first derivative) is made to vanish. The parameter represents the proper
43
44
Black Holes, Information, and the String Theory Revolution
distance of the cutoff point to the horizon. Physically we are introducing
a perfectly reflecting mirror just outside the horizon at a distance . Later
we will remove the cutoff by allowing uo → −∞.
It is by no means obvious that a reflecting boundary condition very near
the horizon is a physically reasonable way to regularize the theory. However
it will prove interesting to separate physical quantities into those which are
sensitive to and those which are not. Those things which depend on are
sensitive to the behavior of the physical theory at temperatures of order
1
2π and greater.
Each transverse Fourier mode χk can be thought of as a free 1+1 dimensional quantum field confined to a box. One end of the box is at the
reflecting boundary at u = uo = log . The other wall of the box is provided
by the repulsive potential
V (u) = k 2 exp(2u)
which becomes large when u > −log k. Thus we may approximate the
potential by a second wall at u = u1 = −log k. The total length of the box
depends on k and according to
L(k) = −log( k)
(4.0.2)
For each value of k the field χk can be expanded in mode functions and
creation and annihilation operators according to
χk (u) =
∗
a+ (n, k) fn,k (u) + a− (n, k) fn,k
(u)
(4.0.3)
n
where the mode (n, k) has frequency λ(n, k). The Rindler Hamiltonian is
given by
HR = d2 k n λ(n, k) a† (n, k) a(n, k)
(4.0.4)
=
λ(n,
k)
N
(n,
k)
n
where
N (n, k) = a† (n, k) a(n, k)
(4.0.5)
Thus far the quantization rules are quite conventional. The new and
unusual feature of Rindler quantization, encountered in Chapter 3, is that
Entropy of the Free Quantum Field in Rindler Space
45
we do not identifiy the vacuum with the state annihilated by the a(n, k),
but rather with the thermal density matrix
ρR =
ρR (n, k)
(4.0.6)
n,k
with
ρR (n, k) ∼ exp −2πλ(n, k) a† (n, k) a(n, k)
(4.0.7)
Thus the average occupation number of each mode is
N (n, k) =
1
exp[2πλ(n, k)] − 1
(4.0.8)
These particles constitute the thermal atmosphere.
The reader might wonder what goes wrong if we choose the state which
is annihilated by the a’s. Such a state is not at all invariant under translations of the original Minkowski coordinates Z and T . In fact, a careful
computation of the expectation value of T µν in this state reveals a singular
behavior at the horizon. Certainly this is not a good candidate to represent
the original Minkowski vacuum.
A black hole, on the other hand, is not a translationally invariant system.
One might therefore suppose that the evolution of the horizon might lead
to the Fock space vacuum with no quanta rather than the thermal state.
This however would clearly violate the fourth guiding principle stated in
the introduction: To a freely falling observer, the horizon of a black hole
should in no way appear special. Moreover, the large back reaction on the
gravitational field that would result from the divergent expectation value
of T µν makes it unlikely that this state can exist altogether.
Physical quantities in Rindler space can be divided into those which
are sensitive to the cutoff at and those which are not. As an example of
insensitive quantities, the field correlation functions such as
χ(X, Y, u) χ(X , Y , u ) = T r ρ χ(X, Y, u) χ(X , Y , u ),
(4.0.9)
are found to have smooth limits as → 0, as long as the points (X, Y, u)
and (X , Y , u ) are kept away from the horizon. Therefore such quantities
can be said to decouple from the degrees of freedom within a distance of
the horizon. A much more singular quantity which will be of great concern
in future lectures is the entropy of the vacuum state.
46
Black Holes, Information, and the String Theory Revolution
Since the relevant density matrix has the Maxwell–Boltzmann form, we
can use equations 3.3.16 and 3.3.17 to obtain the entropy. Defining
T r e−βH = Z(β)
(4.0.10)
and using the identity
∂ N ρ ∂N
N =1
ρ log ρ =
we obtain
S = −T r
(4.0.11)
∂ e−N βH ∂N Z(β)N N =1
= +T r βH
e−βH
Z
+ lnZ
(4.0.12)
= β H + lnZ
Defining E = H and F = − β1 logZ we find the usual thermodynamic
identity
S = β(E − F )
(4.0.13)
where we find
Another identity follows from using E = − ∂logZ
∂β
S = −β 2
∂(logZ/β)
∂β
(4.0.14)
The entropy S in equations 4.0.13 and 4.0.14 can be thought of as both
entanglement and thermal entropy in the special case of the Rindler space
density matrix. This is because the effect of integrating over the fields χL
in equation 3.4.28 is to produce the thermal density matrix in equation
3.4.30. Thus the computation of the entropy of Rindler space is reduced to
ordinary thermodynamic methods. For the present case of free fields the
entropy is additive over the modes and can be estimated from the formula
for the thermodynamics of a free 1+1 dimensional scalar field.
To compute the total entropy we begin by replacing the infinite transverse X, Y plane by a finite torus with periodic boundary conditions. This
has the effect of discretizing the values of k. Thus
kX =
2nX π
B
where B is the size of the torus.
,
kY =
2nY π
B
(4.0.15)
Entropy of the Free Quantum Field in Rindler Space
47
The entropy stored in the field χk can be estimated from the entropy
density of a 1+1 dimensional massless free boson at temperature T . A
S
to be given by
standard calculation gives the entropy density L
S
π
=
T
L
3
(4.0.16)
where T is the temperature. Substituting T =
the length L gives the entropy of χk
S(k) =
1
2π
and equation 4.0.2 for
1
|log k |
6
(4.0.17)
To sum over the values of k we use equation 4.0.15 and let B → ∞
ST otal =
B2
24π 2
d2 k |log k |
(4.0.18)
In evaluating equation 4.0.18, the integral must be cut off when k > 1 .
This is because when k = 1 the potential is already large at u = uo so that
the entire contribution of χk is supressed. We find that S is approximately
given by
ST otal ≈
1 B2
96π 2 2
(4.0.19)
From equation 4.0.19 we see two important features of the entropy of
Rindler space. The first is that it is proportional to the transverse area
of the horizon, B 2 . One might have expected it to diverge as the volume
of space, but this is not the case. The entropy is stored in the vicinity of
the stretched horizon and therefore grows only like the area. The second
feature which should alarm us is that the entropy per unit area diverges like
1
2 . As we shall see, the entropy density of the horizon is a physical quantity
whose exact value is known. Nevertheless the divergence in S indicates that
its value is sensitive to the ultraviolet physics at very small length scales.
Further insight into the form of the entropy can be gained by recalling
1
. Furthermore
that the proper temperature T (ρ) is given by T (ρ) = 2πρ
the entropy density of a 3+1 dimensional free scalar field is given by
2
S(T ) = V 2 ζ(4)kB
π
kB T
c
3
= V
2π 2 3
T
45
(4.0.20)
Now consider the entropy stored in a layer of thickness δρ and area B 2 at
48
Black Holes, Information, and the String Theory Revolution
a distance ρ from the horizon
δS(ρ) =
2π 2
45
T 3 (ρ) δρ B 2
(4.0.21)
=
2π 2
1
45 (2πρ)3
δρ B
2
To find the full entropy we integrate with respect to ρ
∞ dρ
B 2 2π 2
S = (2π)
3 45
ρ3
2π 2
(4.0.22)
B2
45
(2π)3 22
=
Now we see that the entropy is mainly found near the horizon because that
is where the temperature gets large.
4.1
Black Hole Evaporation
The discovery of a temperature seen by an accelerated fiducial observer
adds a new dimension to the equivalence principle. We can expect that
identical thermal effects will occur near the horizon of a very massive black
hole. However, in the case of a black hole a new phenomenon can take
place – evaporation. Unlike the Rindler case, the thermal atmosphere is
not absolutely confined by the centrifugal potential in equation 2.0.7. The
particles of the thermal atmosphere will gradually leak through the barrier
and carry off energy in the form of thermal radiation. A good qualitative
understanding of the process can be obtained from the Rindler quantum
field theory in equation 2.1.22 by observing two facts:
1) The Rindler time ω is related to the Schwarzschild time t by the equation
ω =
t
4M G
(4.1.23)
Thus a field quantum with Rindler frequency νR is seen by the distant
Schwarzschild observer to have a red shifted frequency ν
ν =
νR
4M G
(4.1.24)
The implication of this fact is that the temperature of the thermal atmosphere is reckoned to be red shifted also. Thus the temperature as
Entropy of the Free Quantum Field in Rindler Space
49
seen by the distant observer is
1
1
1
×
=
,
(4.1.25)
2π
4M G
8πM G
a form first calculated by Stephen Hawking.
2) The centrifugal barrier which is described in the Rindler theory by the
potential k 2 exp(2u) is modified at distances r ≈ 3M G as in Figure 2.1.
In particular the maximum value that V takes on for angular momentum
zero is
1
27
(4.1.26)
Vmax ( = 0) =
2
1024 M G2
T =
1/2
√
3 3
Any s-wave quanta with frequencies of order (Vmax )
= 32MG
or
greater will easily escape the barrier. Since the average energies of
massless particles in thermal equilibrium at temperature T is of course
of order T , equation 4.1.25 indicates that some of the s-wave particles
will easily escape to infinity. Unless the black hole is kept in equilibrium
by incoming radiation it will lose energy to its surroundings.
Particles of angular momenta higher than s-waves cannot easily escape
because the potential barrier is higher than the thermal scale. The black
hole is like a slightly leaky cavity containing thermal radiation. Most quanta
in the thermal atmosphere have high angular momenta and reflect off the
walls of the cavity. A small fraction of the particles carry very low angular
momenta. For these particles, the walls are semi-transparent and the cavity
slowly radiates its energy. This is the process first discovered by Hawking
and is referred to as Hawking radiation.
The above description of Hawking radiation does not depend in any
essential way on the free field approximation. Indeed it only makes sense
if there are interactions of sufficient strength to keep the system in equilibrium during the course of the evaporation. In fact, most discussions of
Hawking radiation rely in an essential way on the free field approximation,
and ultimately lead to absurd results. At the end of the next lecture, we
will discuss one such absurdity.
50
Black Holes, Information, and the String Theory Revolution
Chapter 5
Thermodynamics of Black Holes
We have seen that a large black hole appears to a distant observer as a
body with temperature
T =
1
8πM G
(5.0.1)
and energy M . It follows thermodynamically that it must also have an
entropy. To find the entropy we use the first law of thermodynamics in the
form
dE = T dS
(5.0.2)
where E, the black hole energy, is replaced by M . Using equation 5.0.1
dM =
1
8πMG
dS
from which we deduce
S = 4πM 2 G
(5.0.3)
The Schwarzschild radius of the black hole is 2M G and the area of the
horizon is 4π(4M 2 G2 ) so that
SBH =
Area
4G
(5.0.4)
This is the famous Bekenstein–Hawking entropy. It is gratifying that it is
proportional to the area of the horizon. This, as we have seen, is where all
the infalling matter accumulates according to external observers. We have
seen in Chapter 4 equation 4.0.19 that the matter fields in the vicinity of
the horizon give rise to an entropy. Presumably this entropy is part of the
entropy of the black hole, but unfortunately it is infinite as → 0. Evidently
something cuts off the modes which are very close to the horizon. To get an
51
52
Black Holes, Information, and the String Theory Revolution
idea of where the cut off must occur, we can require that the contribution
in equation 4.0.19 not exceed the entropy of the black hole SBH
1
96π 2 2
<
∼
1
4G
(5.0.5)
or
√
>
∼
G
15
(5.0.6)
In other words, the cutoff must not be much smaller than the Planck length,
where
the Planck length is given in terms of Newton’s constant as P =
c3 G.
This is of course not surprising. It is widely believed that the nasty
divergences of quantum gravity will somehow be√cut off by some mechanism
when the distance scales become smaller than G.
What is the real meaning of the black hole entropy? According to the
principles stated in the introduction to these lectures, the entropy reflects
the number of microscopically distinct quantum states that are “coarse
grained” into the single macroscopic state that we recognize as
a black
hole. The number of such states is of order exp SB.H. = exp 4πM 2 G .
Another way to express this is through the level density of the black hole
dN
∼ exp 4πM 2 G
dM
(5.0.7)
where dN is the number of distinct quantum states with mass M in the
interval dM .
The entropy of a large black hole is an extensive quantity in the sense
that it is proportional to the horizon area. This suggests that we can
understand the entropy in terms of the local properties of a limiting black
hole of infinite mass and area. The entropy diverges, but the entropy per
unit area is finite. The local geometry of a limiting black hole horizon is of
course Rindler space.
Let us consider the Rindler energy of the horizon. By definition it is
conjugate to the Rindler time ω. Accordingly we write
[ER (M ), ω] = i
(5.0.8)
Here ER is the Rindler energy which is of course the eigenvalue of the
Rindler Hamiltonian. We assume that for a large black hole the Rindler
energy is a function of the mass of the black hole.
Thermodynamics of Black Holes
53
The mass and Schwarzschild time are also conjugate
[M, t] = i
Now use ω =
t
4MG
to obtain
ER (M ),
t
4MG
(5.0.9)
= i
or
[ER (M ), t] = 4M G i
(5.0.10)
Finally, the conjugate character of M and t allows us to write equation
5.0.10 in the form
∂ER
= 4M G
∂M
(5.0.11)
ER = 2M 2 G
(5.0.12)
and
The Rindler energy and the Schwarzschild mass are both just the energy
of the black hole. The Schwarzschild mass is the energy as reckoned by
observers at infinity using t-clocks, while the Rindler energy is the (dimensionless) energy as defined by observers near the horizon using ω-clocks. It
is of interest that the Rindler energy is also extensive. The area density of
Rindler energy is
1
ER
=
A
8πG
(5.0.13)
The Rindler energy and entropy satisfy the first law of thermodynamics
dER =
1
dS
2π
(5.0.14)
1
where 2π
is the Rindler temperature. Thus we see the remarkable fact
that horizons have universal local properties that behave as if a thermal
membrane or stretched horizon with real physical properties were present.
As we have seen, the stretched horizon also radiates like a black body.
The exact rate of evaporation of the black hole is sensitive to many
details, but it can easily be estimated. We first recall that only the very low
angular momentum quanta can escape the barrier. For simplicity, suppose
that only the s-wave quanta get out. The s-wave quanta are described
1
.
in terms of a 1+1 dimensional quantum field at Rindler temperature 2π
In the same units, the barrier height for the s-wave quanta is comparable
54
Black Holes, Information, and the String Theory Revolution
to the temperature. It follows that approximately one quantum per unit
Rindler time will excape. In terms of the Schwarzschild time, the flux of
1
. Furthermore each quantum carries an energy at
quanta is of order MG
1
. The resulting rate
infinity of order the Schwarzschild temperature 8πMG
1
of energy loss is of order M 2 G2 . We call this L, the luminosity. Evidently
energy conservation requires the black hole to lose mass at just this rate
dM
C
= −L = − 2 2
(5.0.15)
dt
M G
where C is a constant of order unity. The constant C depends on details
such as the number of species of particles that can be treated as light enough
to be thermally produced. It is therefore not really constant. When the
mass of the black hole is large and the temperature low, only a few species
of massless particles contribute and C is constant.
If we ignore the mass dependence of C, equation 5.0.15 can be integrated
to find the time that the black hole survives before evaporating to zero mass.
This evaporation time is evidently of order
tevaporation ∼ M 3 G2
(5.0.16)
It is interesting that luminosity in equation 5.0.15 is essentially the
Stephan–Boltzmann law
L ∼ T 4 · Area
(5.0.17)
1
and Area ∼ M 2 G2 in equation 5.0.15 gives equation 5.0.17.
Using T ∼ MG
However the physics is very different from that of a radiating star. In
that case the temperature and size of the system are related in an entirely
different way. The typical wavelength of a photon radiated from the sun is
∼ 10−5 cm, while the radius of the surface of the sun is ∼ 1011 cm. The sun
is for all intents and purposes infinite on the scale of the emitted photon
wavelengths. The black hole on the other hand emits quanta of wavelength
∼ T1 ∼ M G, which is about equal to the Schwarzschild radius. Observing
a black hole by means of its Hawking radiation will always produce a fuzzy
image, unlike the image of the sun.
Chapter 6
Charged Black Holes
There are a variety of ways to generalize the conventional Schwarzschild
black hole. By going to higher dimensions we can consider not only black
holes, but black strings, black membranes, and so forth. Typically black
strings and branes are studied as systems of infinite extent, and therefore
have infinite entropy. For this reason they can store infinite amounts of
information. Higher dimensional Schwarzschild black holes are quite similar
to their four-dimensional counterparts.
Another way to generalize the ordinary black hole is to allow it to carry
gauge charge and/or angular momentum. In this lecture we will describe
the main facts about charged black holes. The most important fact about
them is that they cannot evaporate away completely. They have ground
states with very special and simplifying features.
Thus, let us consider electrically charged black holes. The metric for a
Reissner–Nordstrom black hole is
−1
Q2 G
Q2 G
2M G
2M G
2
+
+
dr2 + r2 dΩ2
dt + 1 −
ds = − 1 −
r
r2
r
r2
(6.0.1)
The electric field is given by the familiar Coulomb law
2
Er = rQ2
Eθ,φ = 0
(6.0.2)
If the electric field is too strong at the horizon, it will cause pair production of electrons, which will discharge the black hole in the same manner
as a nucleus with Z >> 137 is discharged. Generally the horizon occurs
at r ∼ M G, and the threshold field for unsupressed pair production is
E ∼ m2e , where me is the electron mass. Pair production is exponentially
55
56
Black Holes, Information, and the String Theory Revolution
suppressed if
Q
<< m2e
M 2 G2
(6.0.3)
2
or alternativley MQ >> m21G2 .
e
For Q2 > M 2 G the metric in equation 6.0.1 has a time-like singularity
with no horizon to cloak it. Such “naked singularities” indicate a breakdown of classical relativity visible to a distant observer. The question is
not whether objects with Q2 > M 2 G can exist. Clearly they can. The
electron is such an object. The question is whether they can be described
by classical general relativity. Clearly they cannot. Accordingly we restrict
2
2
our attention to the case M 2 > QG or MQ > Q
G . A Reissner–Nordstrom
black hole that saturates this relationship M 2 =
black hole. Thus equation 6.0.3 is satisfied if
Q >>
1
m2e G
Q2
G
is called an extremal
∼ 1044
Black holes with charge >> 1044 can only discharge by exponentially suppressed tunneling processes. For practical purposes we regard them as
stable.
The Reissner–Nordstrom solution has two horizons, an outer one and
an inner one. They are defined by
Q2 G
2M G
+ 2
= 0
(6.0.4)
1−
r±
r±
where r+ (r− ) refers to the outer (inner) horizon:
Q2
r± = M G 1 ± 1 − 2
M G
(6.0.5)
The metric can be rewritten in the form
ds2 = −
(r − r+ )(r − r− ) 2
r2 dr2
+ r2 dΩ2
dt +
2
r
(r − r+ )(r − r− )
2
(6.0.6)
Note that in the extremal limit M 2 = QG the inner and outer horizons
merge at r± = M G.
To examine the geometry near the outer horizon, let us begin by computing the distance from r+ to an arbitrary point r > r+ . Using equation 6.0.6
Charged Black Holes
57
we compute the distance ρ to be
r
dr
(r − r+ )(r − r− )
(6.0.7)
r+ + r− ≡ Σ
r+ − r− ≡ ∆
y ≡ r − Σ2
(6.0.8)
ρ =
We define the following
We find
ρ =
= y2 −
y+ Σ
2
2
y 2 −( ∆
2 )
∆2
4
+
Σ
2
dy
cosh
−1
2 ∆y
(6.0.9)
The radial-time metric is given by
2
y 2 − ∆4
2
2
ds = − 2 dt + dρ
y + Σ2
2
(6.0.10)
Expanding equation 6.0.9 near the horizon r+ one finds
1/2 2r+
∆
ρ ≈ y −
.
2
∆1/2
(6.0.11)
Note that the proper distance becomes infinite for extremal black holes.
For non-extremal black holes, equation 6.0.10 becomes
ds2 ∼
=
∆2
4
4r+
ρ2 dt2 − dρ2
(6.0.12)
∼
= ρ2 dω 2 − dρ2
where
ω ≡
∆
2 t
2r+
(6.0.13)
Evidently the horizon geometry is again well approximated by Rindler
Q
space. The charge density on the horizon is 4πr
2 . Since r+ ∼ M G the
+
58
Black Holes, Information, and the String Theory Revolution
charge density is ∼ 4πMQ2 G2 . Thus for near extremal black holes, the charge
density is of the form
√
M G ∼
1
Q
∼
(6.0.14)
=
2
2
2
4πr+
4πM G
4πM G3/2
For very massive black holes the charge density becomes vanishingly small.
Therefore the local properties of the horizon cannot be distinguished from
those of a Schwarzschild black hole. In particular, the temperature at a
1
. From equation 6.0.13 we can
small distance ρo from the horizon is 2πρ
o
compute the temperature as seen at infinity.
T (∞) =
∆ 1
2 2π
2r+
(6.0.15)
Using
∆ = 2M G 1 −
Q2
M2G
r+ = M G 1 + 1 −
We find
(6.0.16)
Q2
M2G
2
2M G 1 − MQ2 G
T (∞) =
2
2
2
4πM G 1 + 1 − MQ2 G
2
(6.0.17)
As the black hole tends to extremality, the horizon becomes progressively more removed from any fiducial observer. From equation 6.0.9 we
see that as ∆ → 0
ρ → y +
Σ
log(2y) − log∆
2
(6.0.18)
Thus for a fiducial observer at a fixed value of r the horizon recedes to
infinite proper distance as ∆ → 0.
In the limit ∆ → 0 the geometry near the horizon simplifies to the form
2
ρ
2
2
2
2
dω + dρ + r+
dΩ2
(6.0.19)
ds = − r+ sinh
r+
which, although infinitely far from any fiducial observer with r = r+ , is
approximately Rindler.
Charged Black Holes
59
We note from equation 6.0.17 that in the extremal limit the temperature
at infinity tends to zero. The entropy, however, does not tend to zero. This
can be seen in two ways, by focusing either on the region very near the
horizon or the region at infinity. As we have seen, the local properties of
the horizon even in the extreme limit are identical to the Schwarzschild case
1
. Accordingly,
from which we deduce an entropy density 4G
S =
2
πr+
Area
=
= πM 2 G
4G
G
(6.0.20)
We can deduce this result by using the first law together with equation
6.0.17
dM = T dS
(Fixed Q)
to obtain S = Area
4G as a general rule.
The fact that the temperature goes to zero in the extreme limit indicates
that the evaporation
process slows down and does not proceed past the
√
point Q = M G. In other words, the extreme limit can be viewed as the
ground state of the charged black hole. However it is unusual in that the
entropy does not also tend to zero. This indicates that the ground state
is highly degenerate with a degeneracy ∼ eS . Whether this degeneracy
is exact or only approximate can not presently be answered in the general
case. However in certain supersymmetric cases the supersymmetry requires
exact degeneracy.
The metric in equation 6.0.19 for extremal black holes can be written
in a form analogous to equations 1.3.18 and 1.4.21 by introducing a radial
variable
eρ/r+ − 1
R
= ρ/r
r+
e + +1
The metric then takes the form
⎞
⎤
⎡⎛
2 ⎠
2
dτ 2 = ⎣⎝
R2 dω 2 − dR2 ⎦ − r+
dΩ2
2
1− R
r2
(6.0.21)
(6.0.22)
+
Obviously the physics near rR+ → 0 is identical to Rindler space, from which
it follows that the horizon will have the usual properties of temperature,
entropy, and a thermal atmosphere including particles of high angular momenta trapped near the horizon by a centrifugal barrier.
60
Black Holes, Information, and the String Theory Revolution
Although the external geometry of an extreme or near extreme Reissner–
Nordstrom black hole is very smooth with no large curvature, one can nevertheless expect important quantum effects in its structure. To understand
why, consider the fact that as ∆ → 0 the horizon recedes to infinity. Classically, if we drop the smallest amount of energy into the extreme black
hole, the location of the horizon, as measured by its proper distance, jumps
an infinite amount. In other words, the location of the horizon of an extremal black hole is very unstable. Under these circumstances, quantum
fluctuations can be expected to make the location very uncertain. Whether
this effect leads to a lifting of the enormous degeneracy of ground states
or any other physical phenomena is not known at present except in the
supersymmetric case.
Chapter 7
The Stretched Horizon
Thus far our description of the near-horizon region of black holes, or
Rindler space, has been in terms of quantum field theory in a fixed background geometry. But we have already run into a contradiction in applying
quantum field theory, although we didn’t spell it out. The problem arose
in Chapter 4 when we found that the entropy per unit area of the horizon
diverges as the cutoff tends to zero (see equation 4.0.22). That in itself
is not a problem. What makes it a problem is that we later found that
A
. Free quantum
black hole thermodynamics requires the entropy to be 4G
field theory is giving too much entropy in modes very close to the horizon,
where the local temperature diverges. The fact that the entropy is infinite in quantum field theory implies that any quantity that depends on the
finiteness of the entropy will be miscalculated using quantum field theory.
One possibility is that we have overestimated the entropy by assuming
free field theory. Equation 4.0.20 could be modified by interactions. Indeed
that is so, but the effect goes in the wrong direction. The correct entropy
density for a general field theory can always be parameterized by
S(T ) = γ(T ) T 3
where γ(T ) represents the number of “effective” degrees of freedom at temperature T . It is widely accepted and in many cases proven that γ(T ) is
a monotonically increasing function of T . Thus, conventional interactions
are only likely to make things worse. What we need is some new kind of
theory that has the effective number of degrees of freedom going to zero
very close to the horizon. Let’s suppose that ordinary quantum field theory
is adequate down to distance scale . In order that the entropy at distance
greater than not exceed the Bekenstein–Hawking value, we must have the
rough inequality
61
62
Black Holes, Information, and the String Theory Revolution
2
<
∼
G =
2
P
√
Evidently at distances less than G from the horizon the degrees of freedom must be very sparse, or even nonexistent. This leads to the idea that
the mathematical horizon should be replaced by an effective membrane, or
“stretched” horizon at a distance of roughtly one Planck length from the
mathematical horizon.
Stretching the horizon has another benefit. Instead of being light-like, a
system at the stretched horizon is time-like. This means that real dynamics
and evolution can take place on the stretched horizon. As we will see, the
stretched horizon has dynamics of its own that includes such phenomena
as viscosity and electrical conductivity.
To see that the horizon of a black hole has electrical properties, it is
sufficient to study electrodynamics in Rindler space. First let us define the
stretched horizon. The metric is
dτ 2 = ρ2 dω 2 − dρ2 − dx2⊥
(7.0.1)
The stretched horizon is just the surface
ρ = ρo
(7.0.2)
where ρo is a length of order the Planck length.
The action for the electromagnetic field in Rindler space is
√
−g µν στ
g g Fµσ Fντ + j µ Aµ dω dρ d2 x⊥
W =
16π
(7.0.3)
or, substituting the form of the metric
⎡
W =
⎛
˙ ⎢ 1 ⎜ A + ∇φ
⎣
⎝
8π
ρ
⎞
2
×A
− ρ ∇
⎤
2⎟
⎥
⎠ + j · A⎦ dω dρ d2 x⊥
(7.0.4)
˙
∂A
where A means ∂ω and φ = −A0 , and j is a conserved current in the usual
sense ∂µ j µ = 0. As usual
˙
= −∇φ
−A
E
= ∇
×A
B
The Stretched Horizon
With these definitions, the action becomes
⎡
⎞
⎛ 2
⎤
E
2
⎢ 1 ⎜
⎟
⎥
W =
− ρ B
⎠ + j · A⎦ dω dρ d2 x⊥
⎣
⎝
8π
ρ
63
(7.0.5)
and the Maxwell equations are
1
ρ
˙ − ∇
× (ρB) = −4πj
E
×E
= 0
Ḃ + ∇
(7.0.6)
1
ρ
·
∇
E
= 4πj 0
·B
= 0
∇
We begin by considering electrostatics. By electrostatics we mean the
study of fields due to stationary or slowly moving charges placed outside
the horizon. Since the charges are slowly moving in Rindler coordinates,
it means that they are experiencing proper acceleration. We also assume
all length scales associated with the charges are much larger than ρo . In
particular, the distance of the charges from the stretched horizon is macroscopic.
The surface charge density on the stretched horizon is easily defined.
It is just the component of the electric field perpendicular to the stretched
horizon, or more precisely
1
Eρ σ = 4πρ
ρ=ρo
1
= − 4πρ
∂ρ φ
(7.0.7)
ρ=ρo
Working in the Coulomb gauge, the third expression in equation 7.0.6 becomes
= −∇
· 1 ∇φ
· 1E
= 0
(7.0.8)
∇
ρ
ρ
near the stretched horizon. Thus
∂ρ2 φ −
1
∂ρ φ = −∇2⊥ φ
ρ
(7.0.9)
64
Black Holes, Information, and the String Theory Revolution
We can solve this equation near the horizon by the ansatz φ ∼ ρα . The
right hand side will be smaller than the left hand side by 2 powers of ρ and
can therefore be ignored. We easily find that α = 0 or α = 2. Thus we
assume
φ = F (x⊥ ) + ρ2 G(x⊥ ) + terms higher order in ρ
(7.0.10)
Plugging equation 7.0.10 into equation 7.0.9 and evaluating at ρ = ρo gives
∇2⊥ F + ρ2o ∇2⊥ G = 0
(7.0.11)
If ρo is much smaller than all other length scales, then equation 7.0.11 is
simplified to
∇2⊥ F = 0
(7.0.12)
A similar equation can also be derived for the finite mass black hole.
Since the black hole horizon is compact, equation 7.0.12 proves that
φ = constant on the horizon. This is an interesting result, which proves
that the horizon behaves like an electrical conductor.
We can easily identify the surface current density. Taking the time
derivative of equation 7.0.7 and using Maxwell’s equations 7.0.6 gives
4π σ̇ =
1
Ėρ =
ρo
× ρB
∇
(7.0.13)
ρ
Evidently this is a continuity equation if we define:
4π jx = −ρ By
(7.0.14)
4π jy = ρ Bx
Now let us consider an electromagnetic wave propagating toward the
stretched horizon along the ρ axis. From Maxwell’s equations we obtain
Ḃx = ∂ρ Ey
Ḃy = −∂ρ Ex
1
ρ Ėx
1
ρ Ėy
(7.0.15)
= −∂ρ (ρ By )
= ∂ρ (ρ Bx )
The Stretched Horizon
65
To make the equation more familiar, we can redefine the magnetic field
= β
ρB
(7.0.16)
and use tortoise coordinates
u = log ρ
Equation 7.0.15 then becomes
β̇x = ∂u Ey
β̇y = −∂u Ex
(7.0.17)
Ėx = ∂u βy
Ėy = −∂u βx
The mathematical equations allow solutions in which the wave propagates in either direction along the u-axis. However the physics only makes
sense for waves propagating toward the horizon from outside the black hole.
For such waves, the Maxwell equations 7.0.17 give
βx = Ey
(7.0.18)
βy = −Ex
or from equation 7.0.14
jx =
1
4π
Ex
jy =
1
4π
Ey
Evidently the horizon is an ohmic conductor with a resistivity of 4π.
That corresponds to a surface resistance of 377Ω/square.∗ For example, if
a circuit is constructed as in Figure 7.1, a current will flow precisely as if
the horizon were a conducting surface.
As a last example let us consider dropping a charged point particle
into the horizon. Since the horizon is just ordinary flat space, one might
conclude that the point charge just asymptotically approaches the horizon
∗ The
unit Ω/square is not a misprint. The resistance of a two-dimensional resistor is
scale invariant and only depends on the shape.
66
Black Holes, Information, and the String Theory Revolution
A
A
Horizon
Fig. 7.1
Battery, ammeter attached to horizon
with the transverse charge density remaining point-like. However this is
not at all what happens on the stretched horizon. This process is shown
in Figure 7.2. Without loss of generality, we can take the charge to be
at rest at position zo in Minkowski coordinates. To compute the surface
charge density on the stretched horizon, we need to determine the field
component Eρ . The calculation is easy because at any given time the
Rindler coordinates are related to the Minkowski coordinates by a boost
along the z-axis. Since the component of electric field along the boost
direction is invariant, we can write the standard Coulomb field
Eρ = Ez
=
=
e (z−zo )
[(z−zo )2 + x2⊥ ]3/2
e (ρ cosh ω − zo )
(7.0.19)
3/2
[(ρ cosh ω − zo )2 + x2⊥ ]
Using 4π σ =
Eρ ρ we find
ρo
σ =
ρo cosh ω − zo
e
3/2
4πρo
2
(ρo cosh ω − zo ) + x2⊥
(7.0.20)
Now let’s consider the surface density for large Rindler time.
σ =
e
ρo e ω
4πρo [ρ2 e2ω + x2 ]3/2
o
⊥
(7.0.21)
The Stretched Horizon
67
H
or
iz
on
T
Freely falling charge
Z0
Z
Stretched
horizon
Fig. 7.2
Charge falling past the stretched horizon
To better understand this expression, it is convenient to rescale x⊥ using
x⊥ = eω y⊥ to obtain
σ =
e−2ω
e
4π (ρ2 + y 2 )3/2
o
⊥
(7.0.22)
It is evident that the charge spreads out at an exponential rate with Rindler
time. For a real black hole, it would spread over the horizon in a time
ω = log (Rs − ρo ) = log (2M G − ρo )
or in terms of the Schwarzschild time
t = 4M Gω = 4M G log (2M G − ρo )
68
Black Holes, Information, and the String Theory Revolution
The exponential spreading of the charge is characteristic of an Ohm’s
law conductor. To see this we use Ohm’s law j = conductivity E. Taking
· j ∼ ∇
·E
∼ σ. Now use the continuity equation get
the divergence gives ∇
the relation σ̇ ∼ −σ. Evidently the surface charge density will exponentially
decrease, and conservation of charge will cause it to spread exponentially.
Thus we see that the horizon has the properties of a more or less conventional hot conducting membrane. In addition to temperature, entropy,
and energy, it exhibits dissipative effects such as electrical resistivity and
viscosity. The surprising and puzzling thing is that they are completely
unnoticed by a freely falling observer who falls through the horizon!
Chapter 8
The Laws of Nature
In this chapter, we want to review three fundamental laws of nature
whose compatibility has been challenged. These laws are:
1)
The principle of information conservation
2)
The equivalence principle
3)
The quantum xerox principle
8.1
Information Conservation
In both classical and quantum mechanics there is a very precise sense in
which information is never lost from a closed isolated system. In classical
physics the principle is embodied in Liouville’s theorem: the conservation
of phase space volume. If we begin following a system with some limited
knowledge of its exact state, we might represent this by specifying an initial
region Γ(0) in the system’s phase space. The region Γ(0) has a volume VΓ
in the phase space.
Now we let the system evolve. The region Γ(0) = Γ evolves into the
region Γ(t). Liouville’s theorem tells us that the volume of Γ(t) is exactly
the same as that of Γ. In this sense the amount of information is conserved.
In a practical sense, information is lost because for most cases of interest
the region Γ becomes very complicated like a fractal, and if we coarse grain
the phase space, it will appear that Γ is growing. As a definition of coarse
graining, if one takes every point in the phase space and surrounds it by
solid spheres of fixed volume, the union of those spheres is the “coarse
grained” volume of phase space, which indeed grows. This is the origin of
the second law of thermodynamics. This is illustrated in Figure 8.1.
In quantum mechanics, the conservation of information is expressed as
69
70
Black Holes, Information, and the String Theory Revolution
Fig. 8.1
Evolution of a fixed volume in phase space
the unitarity of the S-matrix. If we again approach a system with limited
knowledge, we might express this by a projection operator onto a subspace,
P , instead of a definite state. The analog of the phase space volume is the
dimensionality of the subspace
N = TrP
The unitarity of the time evolution insures that N is conserved with time.
A more refined definition of information is provided by the concept of
entropy. Suppose that instead of specifying a region Γ in phase space, we
instead specify a probability density ρ(p, q) in phase space. A generalization
of the volume is given by the exponential of the entropy VΓ → exp S, where
S = −
dp dq ρ(p, q) log ρ(p, q)
(8.1.1)
It is easy to check that if ρ = V1Γ inside Γ and zero outside, then S = log VΓ .
Similarly, for quantum mechanics the sharp projector P can be replaced
by a density matrix ρ. In this case the fine grained or Von Neumann entropy
is
S = −T r ρ log ρ
(8.1.2)
For the case ρ = T Pr P the entropy is log N . Thus the entropy is an estimate of the logarithm of the number of quantum states that make up the
initial ensemble. In both quantum mechanics and classical mechanics the
equations of motion insure the exact conservation of S for a closed, isolated
system.
The Laws of Nature
8.2
71
Entanglement Entropy
In classical physics, the only reason for introducing a phase space probability is a lack of detailed knowledge of the state. In quantum mechanics,
there is another reason, entanglement. Entanglement refers to quantum
correlations between the system under investigation and a second system.
More precisely, it involves separating a system into two or more subsystems.
Consider a composite system composed of 2 subsystems A and B. The
subsystem A(B) is described by some complete set of commuting observables α (β). Let us assume that the composite system is in a pure state
with wave function Ψ(α, β). Consider now the subsystems separately. All
measurements performed on A(B) are describable in terms of a density
matrix ρA (ρB ).
Ψ∗ (α, β) Ψ(α , β)
(ρA )αα =
β
(ρB )ββ =
∗
α Ψ (α, β) Ψ(α, β )
(8.2.3)
The fact that a subsystem is described by a density matrix and not
a pure state may not be due to any lack of knowledge of the state of
the composite system. Even in the case of a pure state, the constituent
subsystems are generally not described by pure states. The result is an
“entanglement entropy” for the subsystems.
Let us consider some properties of the density matrix. For definiteness,
consider ρA , but we could equally well focus on ρB .
1) The density matrix is Hermitian
(ρA )αα = (ρA )∗α α
(8.2.4)
2) The density matrix is positive semidefinite. This means its eigenvalues
are all either positive or zero.
3) The density matrix is normalized to 1.
Trρ = 1
(8.2.5)
It follows that all the eigenvalues are between zero and one. If one of the
eigenvalues of ρA is equal to 1, all the others must vanish. In this case
the subsystem A is in a pure state. This only happens if the composite
wave function factorizes
Ψ(α, β) = ψA (α) ψB (β)
(8.2.6)
72
Black Holes, Information, and the String Theory Revolution
In this case B is also in a pure state.
4) The nonzero eigenvalues of ρA and ρB are equal if the composite system
is in a pure state. To prove this, we start with the eigenvalue equation
for ρA . Call φ the eigenvector of ρA . Then the eigenvalue condition is
βα
Ψ∗ (α, β) Ψ(α , β) φ(α ) = λ φ(α)
We assume λ = 0. Now we define a candidate eigenvector of ρB by
χ(β ) ≡
α
Ψ∗ (α , β ) φ∗ (α ).
Then
β
(ρB )ββ χ(β ) =
=
αα β αβ Ψ∗ (α, β) Ψ(α, β ) χ(β )
Ψ∗ (α, β) Ψ(α, β ) Ψ∗ (α , β ) φ∗ (α )
= λ
α
Ψ∗ (α, β) φ∗ (α)
= λ χ(β)
Thus χ(β) is an eigenvector of ρB with eigenvalue λ.
From the equality of the non-vanishing eigenvalues of ρA and ρB an
important property of entanglement entropy follows:
SA = −T r ρA log ρA = SB
(8.2.7)
Thus we can just refer to the entanglement entropy as SE .
The equality of SA and SB is only true if the combined state is pure.
In that case, the entropy of the composite system vanishes
SA+B = 0
Evidently entropy is not additive in general.
Next, let us consider a large system Σ that is composed of many similar
small subsystems σi . Let us suppose the subsystems weakly interact, and
the entire system is in a pure state with total energy E. Each subsystem
on the average will have energy .
It is a general property of most complex interacting systems that the
density matrix of a small subsystem will be thermal
ρi =
e−β Hi
Zi
(8.2.8)
The Laws of Nature
73
where Hi is the energy of the subsystem. The thermal density matrix maximizes the entropy for a given average energy . In general large subsystems
or the entire system will not be thermal. In fact, we will assume that the
entire system Σ is in a pure state with vanishing entropy.
The coarse grained or thermal entropy of the composite system is defined
to be the sum of the entropies of the small subsystems
ST hermal =
Si
(8.2.9)
i
By definition it is additive. The coarse grained entropy is what we usually
think of in the context of thermodynamics. It is not conserved. To see
why, suppose we start with the subsystems in a product state with no
corelations. The entropy of each subsystem Si as well as the entropy of
the whole system Σ given by σΣ , and the coarse grained entropy of Σ all
vanish.
Now the subsystems interact. The wave function develops correlations,
meaning that it now fails to factorize. In this case, the subsystem entropies
become nonzero
Si = 0
and the coarse grained entropy also becomes nonzero
ST hermal =
i
Si = 0
However, the “fine grained” entropy of Σ is exactly conserved and therefore
remains zero.
Let us consider an arbitrary subsystem Σ1 of Σ which may consist of
one, many, or all of the subsystems σi . Typically the fine grained entropy of
Σ1 is defined as the entanglement entropy S(Σ1 ) of Σ1 with the remaining
subsystem Σ − Σ1 . This will always be less than the coarse grained entropy
of Σ1
ST hermal (Σ1 ) > S(Σ1 )
(8.2.10)
For example, as Σ1 approaches Σ, the fine grained entropy S(Σ1 ) will tend
to zero.
Another concept that we can now make precise is the information in a
subsystem. The information can be defined by
I = ST hermal − S
(8.2.11)
74
Black Holes, Information, and the String Theory Revolution
Often the coarse grained entropy is the thermal entropy of the system,
so that the information is the difference between coarse grained and fine
grained entropy.
Since typical small subsystems have a thermal density matrix, the information in a small subsystem vanishes. At the opposite extreme the
information of the combined system Σ is just its total thermal entropy. It
can be thought of as the hidden subtle correlations between subsystems
that make the state of Σ pure.
How much information are in a moderately sized subsystem? One
might think that the information smoothly varies from zero (for the σi )
to SCoarse Grained (for Σ). However, this is not so. What actually happens is that for subsystems smaller than about 1/2 of the total system, the
information is negligible.
Entropy and information are naturally measured in “bits”. A bit is the
entropy of a two state system if nothing is known[2]. The numerical value
of a bit is log 2. Typically for subsystems less than half the size of Σ the
information is smaller than 1 bit. The subsystem 12 Σ has about 1 bit of
information. Thus for Σ1 < 12 Σ
S(Σ1 ) ∼
= ST hermal (Σ1 )
I(Σ1 ) ≈ 0
Next consider a subsystem with Σ1 > 12 Σ. How much information does
it have? To compute it, we use two facts:
S(Σ − Σ1 ) = S(Σ1 )
S(Σ − Σ1 ) ∼
= ST hermal (Σ − Σ1 )
(8.2.12)
Thus
S(Σ1 ) ∼
= ST hermal (Σ − Σ1 )
(8.2.13)
The coarse grained entropy of Σ − Σ1 will be of order (1 − f ) ST hermal (Σ),
where f is the fraction of the total degrees of freedom contained in Σ1 .
Thus, for Σ1 < 12 Σ the information in Σ1 is essentially zero. However for
The Laws of Nature
75
Σ1 > 12 Σ we get the information to be
I(Σ1 ) = ST hermal (Σ1 ) − S(Σ1 )
= f Sthermal (Σ) − (1 − f ) ST hermal (Σ)
(8.2.14)
= ( 2 f − 1 ) ST hermal (Σ)
We will eventually be interested in the information emitted by a black
hole when it evaporates. For now let’s consider a conventional system which
is described by known laws of physics. Consider a box with perfectly reflecting walls. Inside the box we have a bomb which can explode and fill
the box with radiation. The box has a small hole that allows the thermal
radiation to slowly leak out. The entire system Σ consists of the subsystem B that includes everything in the box. The subsystem A consists of
everyting outside of the box, in this case, outgoing photons.
Initially the bomb is in its ground state, and B has vanishing entropy.
When the bomb explodes, it fills the box with thermal radiation. The
thermal, or coarse grained, entropy of the box increases, but its fine grained
entropy does not. Furthermore, no photons have yet escaped, so S(A) = 0
at this time.
ST hermal (B) = 0
S(B) = 0
(8.2.15)
S(A) = 0
Next, photons slowly leak out. The result is that the interior and exterior of the box become entangled. The entanglement entropy, which is
equal for A and B, begins to increase. The thermal entropy in the box
decreases:
SEntanglement = 0
ST hermal (B) = 0
(8.2.16)
ST hermal (A) = 0
Eventually, all of the photons escape the box. The thermal or coarse
grained entropy as well as the fine grained entropy in the box tends to zero.
The box is in a pure state; its ground state.
76
Black Holes, Information, and the String Theory Revolution
S (Σ 1 )
Pure state
f
1
1
-
2
SThermal (Σ 1 )
Additive coarse-grained
entropy
1
f
1
-
2
I (Σ 1 )
1
-
1
f
2
Fig. 8.2 Top: Von Neumann entropy of Σ1 vs fraction f. Middle: coarse grained
entropy of Σ1 vs fraction f. Bottom: information vs fraction of total degrees of
freedom in Σ1
At this time, the thermal or coarse grained entropy of the exterior radiation has increased to its final value. The second law of thermodynamics
insures that ST hermal (A) is larger than ST hermal (B) just after the explosion. But the fine grained entropy of A must vanish, since the entanglement
has gone to zero.
The Laws of Nature
77
The actual entanglement entropy must be less than the thermal entropy
of A or B. Thus a plot of the various entropies looks like Figure 8.3. Note
that the point at which ST hermal (A) = ST hermal (B) defines the time at
which the information in the outside radiation begins to grow. Before that
point, a good deal of energy has escaped, but no information. Roughly the
point where information appears outside of the box is the point where half
of the final entropy of the photons has emerged.
It is useful to define this time at which information begins to emerge.
This time is called the information retention time. It is the amount of time
that it takes to retrieve a single bit of information about the initial state of
the box.
Thus we see how information conservation works for a conventional
quantum system. The consequence of this principle is the final radiation
field outside the box must be in a pure state. However, this does not mean
that localized regions containing a small fraction of the photons cannot be
extremely thermal. They typically carry negligible information.
The description of the evolution of the various kinds of entropy follow
from very general principles. Thus we regard the conservation of information in black hole evaporation as a fundamental law of nature. Note that
it applies to observations made from outside the black hole.
8.3
Equivalence Principle
A second law of nature concerns the nature of gravitation. In its simplest form the equivalence principle says that a gravitational field is locally
equivalent to an accelerated frame. More exactly, it says that a freely falling
observer or system will not experience the effects of gravity except through
the tidal forces, or equivalently, the curvature components. We have seen
that the magnitude of the curvature components at the horizon are small
and tend to zero as the mass and radius of the black hole increase. The
curvature typically satisfies
R ∼
1
(MG)2
Any freely falling system of size much smaller than M G will not be distorted
or otherwise disrupted by the presence of the horizon.
78
Black Holes, Information, and the String Theory Revolution
S Thermal
A (External)
B
Information
retention
time
(Box)
t
Information
p
entro
ent
m
e
l
ng
Enta
y
Information
retention
time
t
Fig. 8.3 Top: evolution of the thermal entropies of box and exterior. Bottom:
evolution of entanglement entropy and information
The Laws of Nature
79
The equivalence principle requires the horizon of a very large black hole
to have the same effects on a freely falling observer as the horizon of Rindler
space has; namely, no effect at all.
8.4
Quantum Xerox Principle
The third law of nature that plays an important role in the next lecture
is the impossibility of faithfully duplicating quantum information. What
it says is that a particular kind of apparatus cannot be built. We call it
a Quantum Xerox Machine.∗ It is a machine into which any system can
be inserted, and which will copy that system, producing the original and a
duplicate. To see why such a system is not possible, imagine that we insert
a spin in the input port as in Figure 8.4. If the spin is in the up state with
In
Fig. 8.4
Out
Schematic diagram of quantum Xerox machine
respect to the z-axis, it is duplicated
|↑ → |↑ |↑
(8.4.17)
Similarly, if it is in the down state, it is duplicated
|↓ → |↓ |↓
(8.4.18)
Now suppose that the spin is inserted with its polarization along the x-axis,
i.e.
1
√ (|↑ + |↓ )
2
∗ The
quantum Xerox principle is sometimes called the no-cloning principle
(8.4.19)
80
Black Holes, Information, and the String Theory Revolution
The general principles of quantum mechanics require the state to evolve
linearly. Thus from equations 8.4.17 and 8.4.18
1
1
√ (|↑ + |↓ ) → √ (|↑ |↑ + |↓ |↓ )
2
2
(8.4.20)
On the other hand, a true quantum Xerox machine is required to duplicate
the spin in equation 8.4.19
√1
2
=
(|↑ + |↓ ) →
1
2
|↑ |↑ +
1
2
√1
2
(|↑ + |↓ )
|↑ |↓ +
1
2
√1
2
(|↑ + |↓ )
|↓ |↑ +
(8.4.21)
1
2
|↓ |↓
The state in equation 8.4.20 is obviously not the same as that in equation
8.4.21. Thus we see that the principle of linearity forbids the existence of
quantum Xerox machines. If we could construct Xeroxed quantum states,
we would be able to violate the Heisenberg uncertainty principle by a set
of measurements on those states.
Chapter 9
The Puzzle of Information
Conservation in Black Hole
Environments
In 1976 Hawking raised the question of whether information is lost in
the process of formation and evaporation of black holes. By information
loss, Hawking did not mean the practical loss of information such as would
occur in the bomb-in-the-box experiment in Chapter 8. He meant that in
a fine grained sense, information would be lost. In other words, the first
of the laws of nature described in the Introduction would be violated. The
argument was simple and persuasive. It was based on the only available tool
of that time, namely local quantum field theory in the fixed background of a
black hole. Although Hawking’s conclusion is undoubtedly wrong, it played
a central role in replacing the old ideas of locality with a new paradigm.
As we have already seen, quantum field theory has a serious defect when
it comes to describing systems with horizons. It gives rise to an infinite
entropy density on the horizon, instead of the correct Bekenstein–Hawking
c3
. As we will see, quantum field theory must be replaced with an
value of 4G
entirely new paradigm in which the concept of locality is radically altered.
To state the problem, let’s begin with a black hole that has been formed
during a collapse. The Penrose diagram is shown in Figure 9.1. Following
Hawking, we think of the geometry as a background on which we can formulate quantum field theory. Let us concentrate on a theory of massless
particles.
According to the principles of quantum mechanics, the evolution of an
initial state |ψin is governed by a unitary S-matrix, so that the final state
is given by
|ψout = S |ψin
(9.0.1)
One way of stating the principle of information conservation is through the
unitarity of S. The point is that a unitary matrix has an inverse, so that
81
82
Black Holes, Information, and the String Theory Revolution
Singularity
H
or
iz
on
+
Collapsing
energy
-
Fig. 9.1
Penrose diagram of black hole formed by collapse
in principle the initial state can be recovered from the final state
|ψin = S † |ψout
(9.0.2)
Now let us consider the S-matrix in a black hole background. The
Hilbert space of initial states hin is clearly associated with quanta coming
in from I − . These incoming particles will interact and scatter by means of
Feynman diagrams in the black hole background. It is evident from Figure
9.2 that some of the final particles will escape to I + and some will be
lost behind the horizon. Evidently the final Hilbert space, hout is a tensor
product of the states on I+ and those at the singularity S. Thus
hin = hI−
hout = hI+ ⊗ hS
(9.0.3)
In arguing that the final Hilbert space is a tensor product, Hawking
The Puzzle of Information Conservation in Black Hole Environments
83
S
+
-
Fig. 9.2
Feynman diagrams in black hole background
relied on an important property of quantum field theory: locality. Since
every point on the singularity S is space-like relative to every point on I + ,
the operators on S all commute with those on I + . This was central to
Hawking’s analysis.
Now suppose that the final state is given by an S-matrix which connects
hin and hout as in equation 9.0.1. Let us consider the description of the
final state from the viewpoint of the observers at I + . Since they have no
access to S, all experiments on the outgoing particles are described by a
density matrix
ρout = T rsingularity |ψout
ψout |
(9.0.4)
where T rsingularity means a trace over the states on the singularity. Thus, in
general the observer external to the black hole will see a state characterized
by a density matrix.
84
Black Holes, Information, and the String Theory Revolution
A simple example might involve a pair of correlated particles which are
thrown in from I − . By correlated, we mean that the two particles wave
function is entangled. If one particle ends up on S and the other ends up
on I + , then the final state will be entangled. In this case the state on I +
will not be pure.
Hawking went on to make arguments that the purity of the state would
not be restored if the black hole evaporated. In fact, the only possibilities
would seem to be that either information is lost during the entire process
of formation and evaporation, or the information is restored to the outside
world at the very end of the evaporation process, when the singularity is
“exposed” at Planckian temperature.
However, we have seen in Chapter 8 that the maximum amount of
information that can be hidden in a system is its entropy. Once the entropy
of the black hole has evaporated past 12 its original value, it must begin to
come out in the emitted radiation. This is a fundamental law of quantum
mechanics. By the time the black hole has small mass and entropy, the
entanglement entropy of the radiation cannot be larger than the black hole’s
remaining entropy. Thus, even if all information were emitted at the very
end of the evaporation process, a law of nature would be violated from
the viewpoint of the external observer. The situation is even worse if the
information is not emitted at all.
A final possibility that was advocated by some authors is that black
holes never completely evaporate. Instead they end their lives as stable
Planck-mass remnants that contain all the lost information. Obviously
such remnants would have to have an enormous, or even infinite entropy.
Such remnants would be extremely pathological, and we will not pursue
that line further.
9.1
A Brick Wall?
There are two more possibilities worth pointing out. One is that the
horizon is not penetrable. In other words, from the viewpoints of an infalling system, the horizon bounces everything out. A freely falling observer
would encounter a “brick wall” just above the horizon.
The reason that this was never seriously entertained, especially by relativists, is that it badly violates the equivalence principle. Since the near
horizon region of a Schwarzschild black hole is essentially flat space-time,
The Puzzle of Information Conservation in Black Hole Environments
85
any violent disturbance to a freely falling system would violate the second
law of nature in the Introduction. Even more convincing is the fact that
the horizon of a black hole formed by a light-like shell forms before the shell
gets to the center (see Figures 1.10 and 1.12).
Finally, the quantum Xerox principle closes out the last possibility. The
information conservation principle requires all information to be returned
to the outside in Hawking radiation. The equivalence principle, on the
other hand, requires information to freely pass through the horizon. The
quantum Xerox principle precludes both happening. In other words, the
horizon cannot duplicate information, and send one copy into the black hole
while sending a second copy out. Evidently we have come to an impasse. It
seems that some law of nature must break down, at least for some observer.
9.2
Black Hole Complementarity
In its simplest form, black hole complementarity just says that no observer ever witnesses a violation of a law of nature. Thus, for an external
observer it says:
A black hole is a complex system whose entropy is a measure of its
capacity to store information. It tells us that the entropy is the log of the
number of microstates of the degrees of freedom that make up the black
hole. It does not tell us what those micro-degrees of freedom are, but it
allows us to estimate their number. That number is of order the area of
the horizon in Planck units.
Furthermore, it tells us that the micro-degrees of freedom can absorb,
thermalize, and eventually re-emit all information in the form of Hawking
radiation. At any given time, the fine grained entropy of the radiation field
(due to entanglement) cannot exceed the entropy of the black hole. At the
end of evaporation, all information is carried off in Hawking radiation.
For a freely falling observer, black hole complementarity tells us that
the equivalence principle is respected. This means that as long as the
black hole is much larger than the infalling system, the horizon is seen as
flat featureless space-time. No high temperatures or other anomolies are
encountered.
No obvious contradiction is posed for the external observer, since the
infalling observer cannot send reports from behind the horizon. But a
potential contradiction can occur for the infalling observer.
86
Black Holes, Information, and the String Theory Revolution
Singularity
In
fo
rm
at
io
n
M
es
sa
g
e
C
B
A
Fig. 9.3
Information exchange from External to infalling observer
Consider a black hole, shown in Figure 9.3 along with an infalling system A. System A is assumed to contain some information. According to
observations done by A, it passes through the horizon without incident.
Next, consider an observer B who hovers above the black hole monitoring the Hawking radiation. According to assumption, the photons recorded
by observer B encode the information carried in by system A. After collecting some information about A (from the Hawking radiation ), observer
B then jumps into the black hole. We don’t actually need observer B to
decode the information. All we really need is a mirror outside the black
hole horizon to reflect the Hawking radiation back into the black hole.
The Puzzle of Information Conservation in Black Hole Environments
87
We now have two copies of the original information carried by A. We
can imagine A sending a signal to observer B so that observer B discovers
the duplicate information at point C in Figure 9.3. Now we have a contradiction, since observer B has discovered a quantum Xeroxing of information
from observer A. If this experiment is possible, then black hole complementarity is not self-consistent. To see why the experiment fails, we need to
remember that no information will be emitted until about 12 the entropy of
the black hole has evaporated. From equation 5.0.16, this takes a time of
order
t∗ ≈ M 3 G2
(9.2.5)
Let us also assume that the observer B hovers above the horizon at a
distance at least of the order of the Planck length P . In other words,
observer B hovers above the stretched horizon. This means that observer
B’s jump off point must correspond to Rindler coordinates satisfying
ω∗ ≥
t∗
4MG
ρ∗ ≥
P
≈ M2 G
(9.2.6)
In terms of the light cone coordinates x± = ρe±ω we have
−
x+
∗ x∗ >
x+
∗
>
≈
P
2
P
exp (ω ∗ )
(9.2.7)
From Figure 9.4 we can see why there might be a problem with the
experiment. Observer B might be forced to the singularity before a message
can arrive. In fact, the singularity is given by
x+ x− = (M G)
2
(9.2.8)
Thus, observer B will hit the singularity at a point with
x−
<
∼
(M G) e−ω
∗
(9.2.9)
The implication is that if A is to send a signal that B can receive, it must
∗
all occur at x− < (M G) e−ω . This means that A has a time of order
∗
∆t ≈ (M G) e−ω to send the message.
88
Black Holes, Information, and the String Theory Revolution
Singularity
VV
VV
VV
+
V VV
- = (MG)2
V
X
X
V
V
VVVV
VVVVVVVV
M
es
sa
ge
H
or
fro
iz
m
on
A
VV
V
A
Fig. 9.4
B gathers
information
and enters
horizon
Information
from A
P
VV
+
X
B
VVVVVVVV
VVVV
V VV
V
V
VV
VV
V
VV
V
X
Resolution of Xerox paradox for observers within horizon
Now, in the classical theory, there are no limits on how much information can be sent in an arbitrarily small time with arbitrarily small energy. However in quantum mechanics, to send a single bit requires at least
one quantum. Since that quantum must be emitted between x− = 0 and
∗
x− = M G e−ω , its energy must satisfy
E >
∗
1
eω
MG
(9.2.10)
From equation 9.2.6 we see that this energy is exponential in the square
M2 G
of the black hole mass (in Planck units) E > eM G . In other words, for
observer A to be able to signal observer B before observer B hits the singularity, the energy carried by observer A must be many orders of magnitude
larger than the black hole mass. It is obvious that A cannot fit into the
horizon, and that the experiment cannot be done.
This example is one of many which show how the constraints of quan-
The Puzzle of Information Conservation in Black Hole Environments
89
tum mechanics combine with those of relativity to forbid violations of the
complementarity principle.
9.3
Baryon Number Violation
The conservation of baryon number is the basis for the incredible stability of ordinary matter. Nevertheless, there are powerful reasons to believe
that baryon number, unlike electric charge, can at best be an approximate
conservation law. The obvious difference between baryon number and electric charge is that baryon number is not the source of a long range gauge
field. Thus it can disappear without some flux having to suddenly change
at infinity.
Consider a typical black hole of stellar mass. It is formed by the collapse
of roughly 1057 nucleons. Its Schwarzschild radius is about 1 kilometer, and
its temperature is 10−8 electron volts. It is far too cold to emit anything
but very low energy photons and gravitons. As it radiates, its temperature
increases, and at some point it is hot enough to emit massive neutrinos and
anti-neutrinos, then electrons, muons, and pions. None of these carry off
baryon number. It can only begin to radiate baryons when its temperature
has increased to about 1 GeV. Using the connection between mass and
temperature in equation 4.0.22, the mass of the black hole at this point is
about 1010 kilograms. This is a tiny fraction of the original black hole mass,
and even if it were to decay into nothing but protons, it could produce
only about 1037 of them. Baryon number must be violated by quantum
gravitational effects.
In fact, most modern theories predict baryon violation by ordinary quantum field theoretic processes. As a simplified example, let’s suppose there
is a heavy scalar particle X which can mediate a transition between an
elementary proton and a prositron, as well as between two positrons, as
in Figure 9.5. Since the X-boson is described by a real field, it cannot
carry any quantum numbers, and the transition evidently violates baryon
conservation. The proton could then decay into a positron and an electronpositron pair. Let’s also assume that the coupling has the usual Yukawa
form
g ψ̄p ψe+ X + ψ̄e+ ψp X
where g is a dimensionless coupling.
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Black Holes, Information, and the String Theory Revolution
e+
e+
X
X
p
e+
Fig. 9.5
Feynman diagrams of p→ Xe+ and e+ → Xe+
If the mass MX is sufficiently large, baryon conservation will be a very
good symmetry at ordinary energies, just as it is in the real world. The
proton will have a lifetime in excess of 1032 years.
Now let us consider what happens when a proton falls into a black
hole. The baryon number is lost, and will not be radiated back out in the
Hawking radiation. The question is: where does the baryon violation take
place? One possible answer is that it occurs when the freely falling proton
encounters very large curvature invariants as the singularity is approached.
From the proton’s viewpoint, there is nothing that would stimulate it to
decay before that.
On the other hand, from the vantage point of the external observer,
the proton encounters enormously high temperatures as it approaches the
horizon. Temperatures higher than MX can certainly excite the proton to
decay. So the external observer will conclude that baryon violation can
take place at the horizon. Who is right?
The answer that black hole complementarity implies is that they are
both right. On the face of it, that sounds absurd. Surely the event of
proton decay takes place in some definite place. For a very large black
hole, the time along the proton trajectory between horizon crossing and
the singularity can be enormous. Thus, it is difficult to understand how
there can be an ambiguity.
The real proton propagating through space-time is not the simple structureless bare proton. The interactions cause it to make virtual transitions
from the bare proton to a state with an X-boson and a positron. The complicated history of the proton is described by Feynman diagrams such as
shown in Figure 9.6. The diagrams make evident that the real proton is
The Puzzle of Information Conservation in Black Hole Environments
e+
X
X
p
p
p
e+
Fig. 9.6
91
X
e+
p
e+
Proton virtual fluctuation into X e+ pair
a superposition of states with different baryon number; in the particular
processes shown in Figure 9.6, the intermediate state has vanishing baryon
number.
Nothing about virtual baryon non-conservation is especially surprising.
As long as the X-boson is sufficiently massive, the rate for real proton decay
will be very small, and the proton will be stable for long times.
What is surprising is that the probability to find the proton in a configuration with vanishing baryon number is not small. This probability is
closely related to the wave function renormalization constant of the proton,
and is of the order
P robability ∼
g2
4π
log MκX
where κ is the cutoff in the field theory. For example, for g ∼ 1, κ of the
order of the Planck mass, and MX of the order 1016 GeV, the probability
that the proton has the “wrong” baryon number is of order unity. This
might seem paradoxical, since the proton is so stable.
The resolution of the paradox is that the proton is continuously making
extremely rapid transitions between baryon number states. The transitions
take place on a time scale of order δt ∼ M1X . Ordinary observations of the
proton do not see these very rapid fluctuations. The quantity that we
normally call baryon number is really the time averaged baryon number
normalized to unity for the proton.
Now let us consider a proton passing through a horizon, as shown in
Figure 9.7. Since the probability that the proton is actually an X, e+
system is not small, it is not unlikely that when it passes the horizon,
its instantaneous baryon number is zero. But it is clear from Figure 9.7
that from the viewpoint of an external observer, this is not a short-lived
intermediate state. A fluctuation that is much too rapid to be seen by a low
energy observer falling with the proton appears to be a real proton decay
lasting to eternity from outside the horizon.
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Black Holes, Information, and the String Theory Revolution
Time resolution
becomes very dense
∆t
∆t
∆t
∆t
∆t
∆t
∆t
t=0
Fig. 9.7
Proton fluctuations while falling through horizon
This, of course, is nothing but the usual time dilation near the horizon.
As a proton or any other system approaches the horizon, internal oscillations or fluctuations appear to indefinitely slow down, so that a short-lived
virtual fluctuation becomes stretched out into a real process.
The process of baryon violation near the horizon should not be totally
surprising to the external observer. To him, the proton is falling into a
−1
from the
region of increasing temperature. At a proper distance MX
horizon, the temperature becomes of order MX . Baryon violating processes
are expected at that temperature.
An interesting question is whether an observer falling with the proton
The Puzzle of Information Conservation in Black Hole Environments
93
can observe the baryon number just before crossing the horizon, and send
a message to the outside world that the proton has not decayed.
The answer is interesting. In order to make an observation while the
proton is in a region of temperature MX the observer must do so very
quickly. In the proton’s frame, the time spent in the hot region before
crossing the horizon is M1X . Thus, to obtain a single bit of information
about the state of the proton, the observer has to probe it with at least one
quantum with energy of order MX . But such an interaction between the
proton and the probe quantum is at high enough energy that it can cause a
baryon violating interaction even from the perspective of the proton’s frame
of reference. Thus the observer cannot measure and report the absence of
baryon violation at the horizon without himself causing it.
It is evident from this example that the key to understanding the enormous discrepancies in the complementary description of events lies in understanding the fluctuations of matter at very high frequencies; frequencies
so high that the ordinary low energy observer has no chance of detecting
them.
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Black Holes, Information, and the String Theory Revolution
Chapter 10
Horizons and the UV/IR Connection
The overriding theme of 20th century physics was the inverse relation
between size and momentum/energy. According to conventional relativistic
quantum mechanics, a size ∆x can be probed with a quantum of energy
E∆x ≈
c
∆x
(10.0.1)
But we already know that this trend is destined to be reversed in the physics
of the 21st century. This can be seen in many ways. Let’s begin with a
traditional attempt to study interactions at length scales smaller than the
Planck scale. According to conventional thinking, what we need to do is to
collide, head on, particles with center of mass energy E∆x . We expect to
discover high energy collision products flying out at all angles. By analyzing
the highest energy fragments, we hope to reconstruct very short distance
events.
The problem with this thinking is that at energies far above the Planck
mass, the collision will create a black hole of mass ∼ E∆x . The interesting
short distance effects that we want to probe will be hidden behind a horizon
of radius
RS ≈
2G
c2
E∆x
and are inaccessible. To make matters worse, the products of collision will
not be high energy particles, but rather low energy Hawking radiation.
The typical Hawking particle has energy ∼ RcS which decreases with the
incident energy. Thus we see that a giant “Super Plancketron” Collider
(SPC) would fail to discover fundamental length scales smaller than the
Planck scale P , no matter how high the energy. In fact, as the energy
95
96
Black Holes, Information, and the String Theory Revolution
increases we would be probing ever larger scales
∆x ∼
2G E
.
c2 c2
(10.0.2)
We might express this in another form as a kind of space-time uncertainty
principle:
∆x ∆t ∼
2G
≈
c4
2
P
(10.0.3)
This is the simplest example of the ultraviolet/infrared connection that will
control the relation between frequency and spatial size. Very high frequency
is related to large size scale.
The UV/IR connection is deeply connected to black hole complementarity. As we saw in the previous lecture, the enormous differences in the
complementary descriptions of matter falling into a black hole are due to the
very different time resolutions available to the complementary observers.
Let’s consider further: what happens to a proton falling into the black
hole? The proton carries some information with it; its charge, particle type,
momentum, location on the horizon through which it falls, etc. From the
viewpoint of the external observer, the proton is falling into an increasingly
hot region. The proton is like a tiny piece of ice thrown into a tub of very
hot water. The only reasonable expectation is that the constituents of the
proton “melt” and diffuse throughout the horizon. In fact, in Chapter 7, we
saw just such a phenomenon involving a charge falling onto the stretched
horizon. More generally, all of the information stored in the incident proton
should quickly be spread over the horizon. On the other hand, the observer
who follows the proton does not see it spread at all as it falls.
The need to reconcile the complementary descriptions gives us an important clue to the behavior of matter at high frequencies. Following the
proton as it falls, the amount of proper time that it has before crossing the
horizon tends to zero as the Schwarzschild time tends to infinity. Call the
proper time ∆τ . Then ∆τ varies with Schwarzschild time like
√
∆t
(10.0.4)
∆τ ∼ 8M G δR e− 4M G .
Thus in order to observer the proton before it crosses the horizon, we have
to do it in a time which is exponentially small as t → ∞. Thus, what we
need in order to be consistent with the thermal spreading of information is
now clear. As the proton is observed over shorter and shorter time intervals,
the region over which it is localized must grow.
Horizons and the UV/IR Connection
97
In fact, if we assume that it grows consistently with the uncertainty
relation in equation 10.0.3 (substituting ∆τ for ∆t) we obtain
∆x ∼
2
P
∆τ
∼
2
P
MG
et/4MG
Thus, if the proton size depends on the time resolution according to equation 10.0.3, it will spread over the horizon exponentially fast.
The idea of information spreading as the resolving time goes to zero is
very unfamiliar, but it is at the heart of complementarity. It is implicit in
the modern idea of the UV/IR connection. Fortunately it is also built into
the mathematical framework of string theory.
98
Black Holes, Information, and the String Theory Revolution
PART 2
Entropy Bounds and Holography
Chapter 11
Entropy Bounds
11.1
Maximum Entropy
Quantum field theory has too many degrees of freedom to consistently
describe a gravitational theory. The main indication that we have seen
of this overabundance of degrees of freedom is the fact that the horizon
entropy-density is infinite in quantum field theory. The divergence arises
from modes very close to the horizon. One might think that this is just an
indication that a more or less conventional ultraviolet regulator is needed
to render the theory consistent. But the divergent horizon entropy is not
an ordinary ultraviolet phenomenon. The modes which account for the
divergence are very close to the horizon and would appear to be ultra short
distance modes. But they also carry very small Rindler energy and therefore
correspond to very long times to the external observer. This is an example
of the weirdness of the Ultraviolet/Infrared connection.
A quantitative measure of the overabundance of degrees of freedom in
QFT is provided by the Holographic Principle. This principle says that
there are vastly fewer degrees of freedom in quantum gravity than in any
QFT even if the QFT is regulated as, for example, it would be in lattice
field theories.
The Holographic Principle is about the counting of quantum states of a
system. We begin by considering a large region of space Γ. For simplicity
we take the region to be a sphere. Now consider the space of states that
describe arbitrary systems that can fit into Γ such that the region outside
Γ is empty space. Our goal is to determine the dimensionality of that
state-space. Let us consider some preliminary examples.
Suppose we are dealing with a lattice of discrete spins. Let the lattice
spacing be a and the volume of Γ be V . The number of spins is then V /a3
101
102
Black Holes, Information, and the String Theory Revolution
and the number of orthogonal states supported in Γ is given by
3
Nstates = 2V /a
(11.1.1)
A second example is a continuum quantum field theory. In this case the
number of quantum states will diverge for obvious reasons. We can limit
the states, for example by requiring the energy density to be no larger than
some bound ρmax . In this case the states can be counted using some concepts from thermodynamics. One begins by computing the thermodynamic
entropy density s as a function of the energy density ρ. The total entropy
is
S = s(ρ)V
(11.1.2)
The total number of states is of order
Nstates ∼ exp S = exp s(ρmax )V
(11.1.3)
In each case the number of distinct states is exponential in the volume V .
This is a very general property of conventional local systems and represents
the fact that the number of independent degrees of freedom is additive in
the volume.
In counting the states of a system the entropy plays a central role. In
general entropy is not really a property of a given system but also involves
one’s state of knowledge of the system. To define entropy we begin with
some restrictions that express what we know, for example, the energy within
certain limits, the angular momentum and whatever else we may know. The
entropy is essentially the logarithm of the number of quantum states that
satisfy the given restrictions.
There is another concept that we will call the maximum entropy. This
is a property of the system. It is the logarithm of the total number of
states. In other words it is the entropy given that we know nothing about
the state of the system. For the spin system the maximum entropy is
Smax =
V
log 2
a3
(11.1.4)
This is typical of the maximum entropy. Whenever it exists it is proportional to the volume. More precisely it is proportional to the number of
simple degrees of freedom that it takes to describe the system.
Let us now consider a system that includes gravity. For definiteness we
will take spacetime to be four-dimensional. Again we focus on a spherical
Entropy Bounds
103
region of space Γ with a boundary ∂Γ. The area of the boundary is A. Suppose we have a thermodynamic system with entropy S that is completely
contained within Γ. The total mass of this system can not exceed the mass
of a black hole of area A or else it will be bigger than the region.
Now imagine collapsing a spherically symmetric light-like shell of matter
with just the right amount of energy so that together with the original mass
it forms a black hole which just fills the region. In other words the area
of the horizon of the black hole is A. This is shown in Figure 11.1. The
Eg
EM
Γ
Fig. 11.1
R
In-moving (zero entropy) spherical shell of photons
result of this process is a system of known entropy, S = A/4G. But now
we can use the second law of thermodynamics to tell us that the original
entropy inside Γ had to be less than or equal to A/4G. In other words the
maximum entropy of a region of space is proportional to its area measured
in Planck units. Thus we see a radical difference between the number of
states of any (regulated) quantum field theory and a theory that includes
gravity.
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Black Holes, Information, and the String Theory Revolution
Space-time depiction of horizon formation
Consider the collapsing spherically symmetric shell of light-like energy depicted in Figure 11.1. As the photonic shell approaches the center, the
horizon forms prior to the actual crossing of the shell. Inside of the shell,
the geometry is Schwarzschild with low curvature (no black hole) prior to
the shell crossing the Schwarzschild radius. However, just outside of the
shell the geometry becomes increasingly curved as the Schwarzschild radius
is approached (see Figure 11.2). The horizon grows until the collapsing
Light beams get more
vertical as the energy
becomes more concentrated
Horizon
is a
light-like
surface
R
Infalling
Shell E γ
Low
curvature
Fig. 11.2
Space-time depiction of radially in-moving shell of photons
shell crosses, and a singularity forms at a later time. The energy of the
infalling photonic shell Eγ has been tuned such that the collapsing shell
crosses the horizon exactly at the radius R in Figure 11.1. However, at
that time the system winds up with entropy
S=
A
4G .
Entropy Bounds
105
Therefore, unless the second law of thermodynamics is untrue, the entropy
of any system is limited by
Smax ≤
A
.
4G
(11.1.5)
The coarse grained volume in phase space cannot decrease, so this “holographic” limit must be satisfied.
Aside: Scale of Entropy Limit
Example: To get some idea of how big a typical system must be in
order to saturate the maximum entropy, consider thermal radiation at
a temperature of 1000◦K, which corresponds to photons of wavelength
∼10−5 cm. The number of photons Nγ in a volume of radius R satis3
fies Nγ ∼ λV3 ∼ R(cm) ⊗ 105 . Entropy is proportional to the number of
photons, and thus one expects
3
S ∝ R(cm) ⊗ 105 .
Compare this with the maximum entropy calculated using the holographic
limit
2
2
R
∼
(11.1.6)
Smax ≈
= R(cm) ⊗ 1033 .
lP lanck
Evidently the maximum entropy will only be saturated for the photon gas
when the radius is huge, R ∼ 1051 cm, considerable larger than the observable universe 1028 cm.
11.2
Entropy on Light-like Surfaces
So far we have considered the entropy that passes through space-like
surfaces. We will see that it is most natural to define holographic entropy
bounds on light-like surfaces[3] as opposed to space-like surfaces. Under
106
Black Holes, Information, and the String Theory Revolution
certain circumstances the entropy bounds of light-like surfaces can be translated to space-like surfaces, but not always. The case described above is
one of those cases where a space-like bound is derivable.
Let us start with an example in asymptotically flat space-time. We assume that flat Minkowski coordinates X + , X − , xi can be defined at asymptotic distances. In this chapter we will revert to the usual convention in
which X + is used as a light cone time variable. We will now define a “lightsheet”. Consider the set of all light rays which lie in the surface X + = X0+
in the limit X − → +∞. In ordinary flat space this congruence of rays
defines a flat three-dimensional light-like surface. In general, they define
a light-like surface called a light sheet. The light sheet will typically have
singular caustic lines, but can be defined in a unique way[4]. When we vary
X0+ the light sheets fill all space-time except for those points that lie behind
black hole horizons.
Now consider a space-time point p. We will assign it light cone coordinates as follows. If it lies on the light sheet X0+ we assign it the value
X + = X0+ . Also if it lies on the light ray which asymptotically has transverse coordinate xi0 we assign it xi = xi0 . The value of X − that we assign
will not matter. The two-dimensional xi plane is called the Screen. Next
assume a black hole passes through the light sheet X0+ . The stretched
X+
Incoming Parallel
Light Rays
X-
xj
Fig. 11.3
Far off screen
x1 and x 2 plane
Light propagating on light-like surface X + = constant
Entropy Bounds
107
Asymptotic
screen
X
+
Image of
Black hole
on screen
X
stic
Cau
-
xj
Foliation
corresponds
to different
+
x slices
Black
Hole
Fig. 11.4
Family of light rays on fixed X + surface in presence of black hole
horizon of the black hole describes a two-dimensional surface in the threedimensional light sheet as shown in Figure 11.4. Each point on the stretched
horizon has unique coordinates X + , xi , as seen in Figure 11.5. More generally if there are several black holes passing through the light sheet we
can map each of their stretched horizons to the screen in a single valued
manner.
Since the entropy of the black hole is equal to 1/4G times the area of the
horizon we can define an entropy density of 1/4G on the stretched horizon.
The mapping to the screen then defines an entropy density in the xi plane,
σ(x). It is a remarkable fact that σ(x) is always less than or equal to 1/4G.
1
we make use of the focusing theorem of general
To prove that σ(x) ≤ 4G
relativity. The focusing theorem depends on the positivity of energy and is
based on the tendency for light to bend around regions of nonzero energy.
Consider a bundle of light rays with cross sectional area α. The light rays
108
Black Holes, Information, and the String Theory Revolution
Map of patch
on stretched
horizon
Patch on
stretched
horizon
Black
hole
Map of
stretched
horizon
Stretched
horizon
Fig. 11.5
Image of “stretched horizon” on asymptotic screen
are parameterized by an affine parameter λ. The focusing theorem says
that
d2 α
≤0
dλ2
(11.2.7)
Consider a bundle of light rays in the light sheet which begin on the
stretched horizon and go off to X − = ∞. Since the light rays defining the
light sheet are parallel in the asymptotic region dα/dλ → 0. The focusing
theorem tells us that as we work back toward the horizon, the area of the
bundle decreases. It follows that the image of a patch of horizon on the
screen is larger than the patch itself. The holographic bound immediately
follows.
σ(x) ≤
1
4G
(11.2.8)
This is a surprising conclusion. No matter how we distribute the black
holes in three-dimensional space, the image of the entropy on the screen
always satisfies the entropy bound equation 11.2.8. An example which
helps clarify how this happens involves two black holes. Suppose we try to
hide one of them behind the other along the X − axis, thus doubling the
entropy density in the x plane. The bending and focusing of light always
acts as in Figures 11.6 to prevent σ(x) from exceeding the bound. These
considerations lead us to the more general conjecture that for any system,
Entropy Bounds
Motion of
second black
hole
109
Image of
second black
hole
CAUSTIC
CAUSTIC
Motion of
second black
hole
Fig. 11.6
Image of
second black
hole
Initial and later motions and images of second black hole
when it is mapped to the screen the entropy density obeys the bound in
equation 11.2.8.
Thus far we have assumed asymptotically flat boundary conditions.
This allowed us to choose the screen so that the light rays forming the
light sheet intersect the screen at right angles. Equivalently da/dλ equals
zero at the screen. We note for future use that the conclusions concerning
the entropy bound would be unchanged if we allowed screens for which the
light rays were diverging as we move outward, i.e. da/dλ > 0. However,
if we attempt to use screens for which the light rays are converging then
the argument fails. This will play an important role in generalizing the
holographic bound to more general geometries.
Aside: Apparent motions
Consider a single point particle external to the black hole undergoing
motions near a caustic. Examine the projection of those motion upon the
screen, demonstrated in Figures 11.7. One sees that due to gravitational
lensing, the image of the particle can move at arbitrarily large speeds!
produce large, rapid
motions on the screen
Small
motions near
caustic
CAUSTIC
Fig. 11.7
CAUSTIC
Initial and later path and image during “slow” motion near caustic
110
11.3
Black Holes, Information, and the String Theory Revolution
Friedman Robertson Walker Geometry
The holographic bound can be generalized to flat F.R.W. geometries,
where it is called the Fischler–Susskind (FS) bound[5] and to more general
geometries by Bousso[6]. First we will review the F.R.W. case. Consider
the general case of d + 1 dimensions. The metric has the form
dτ 2 = dt2 − a(t)2 dxm dxm
(11.3.9)
where the index m runs over the d spatial directions. The function a(t) is
assumed to grow as a power of t.
a(t) = a0 tp
(11.3.10)
Let’s also make the usual simplifying cosmological assumptions of homogeneity. In particular we assume that the spatial entropy density (per unit
d volume) is homogeneous. Later, we will relax these assumptions.
At time t we consider a spherical region Γ of volume V and area A.
The boundary (d − 1)-sphere, ∂Γ, will play the same role as the screen in
the previous discussion. The light sheet is now defined by the backward
light cone formed by light rays that propagate from ∂Γ into the past (See
Figure 11.8).
t
Screen
Past
light
cone
Projection from
screen onto
light cone
t>0
Holographic surface for calculating entropy bound with a spherical surface as the screen
Fig. 11.8
Entropy Bounds
111
As in the previous case the holographic bound applies to the entropy
passing through the light sheet. This bound states that the total entropy
passing through the light sheet does not exceed A/4G. The key to a proof
is again the focusing theorem. We observe that at the screen the area of the
outgoing bundle of light rays is increasing as we go to later times. In other
words the light sheet has positive expansion into the future and negative
expansion into the past. The focusing theorem again tells us that if we map
the entropy of black holes passing through the light sheet to the screen, the
resulting density satisfies the holographic bound. It is believed that the
bound is very general.
It is now easy to see why we concentrate on light sheets instead of spacelike surfaces. Obviously if the spatial entropy density is uniform and we
choose Γ big enough, the entropy will exceed the area. However if Γ is
larger than the particle horizon at time t the light sheet is not a cone, but
rather a truncated cone as in Figure 11.9, which is cut off by the big bang at
t = 0. Thus a portion of the entropy present at time t never passed through
the light sheet. If we only count that portion of the entropy which did pass
through the light sheet, it will scale like the area A. We will return to the
question of space-like bounds after discussing Bousso’s generalization[6] of
the FS bound.
t
Sphere
Past
light
cone
Entropy flux through
light cone bounded
Big
Bang
t=0
Entropy not
bounded
Fig. 11.9
Region of space causally connected to particle horizon
112
Black Holes, Information, and the String Theory Revolution
Test: Is the observed horizon entropy bounded by its area?
We will check whether the observed particle horizon satisfies the entropy
bound:
SHorizon ≤
AHorizon
4G
???
We can check by recognizing that the entropy is primarily given by the
number of black body cosmic background photons, Nγ ≈ 1090 . The proper
size of the horizon is approximately given by 1018 (light) seconds, and the
Planck time is approximately 10−43 seconds. This gives a proper size for the
horizon of about 1061 Planck units. We can use these numbers to compare
the cosmic entropy with the area of the horizon:
SHorizon ≤???
AHorizon
4G
2
1090 ≤ 1061 = 10122
We find this inequality to definitely be true today.
Next, using the F.R.W. geometry, we will determine if the entropy per
horizon-area contained within the particle horizon is increasing or decreasing. Let R Horizon represent the coordinate size of the particle horizon
(Figure 11.10), and d represent the number of spatial dimensions. Let σ be
the entropy volume density, so that
d
S ∼ σ RHorizon
dτ 2 = dt2 − a2 (t)
d
dxj dxj
j=1
This means that the proper size of the horizon is given by a(t)R Horizon . We
want to check whether
d
<?
S ∼ σ RHorizon
(a(t) RHorizon )
4G
d−1
(11.3.11)
An outgoing light ray (null geodesic) which would generate the particle
horizon satisfies dt = a(t)dx, which gives the form for the time dependence
Entropy Bounds
113
t
RHorizon
Past
light
cone
Entropy
flux
Big
Bang
t=0
Particle Horizon
Fig. 11.10
of the size of the particle horizon:
t
RHorizon (t) =
0
dt
a(t )
(11.3.12)
If we assume the form a(t) = ao tp then the particle horizon evolves according to the formula
RHorizon (t) =
t1−p
.
ao
(11.3.13)
Therefore for the entropy bound to continue to be valid, the time dependence must satisfy t(1−p)d < td−1 . This bounds the expansion rate
coefficient to satisfy
p >
1
.
d
(11.3.14)
One sees that if the expansion rate is too slow, then the coordinate volume
will grow faster than the area, and the entropy bound will eventually be
contradicted.
Suppose that the matter in the F.R.W. cosmology satisfies the equation
of state
P
=
wu
(11.3.15)
where P is the pressure and u is the energy density and w is a constant.
Given w and the scale factor for the expansion p, one can use the Einstein
114
Black Holes, Information, and the String Theory Revolution
field equation to calculate a relationship between them:
p=
1
2
>
d (1 + w)
d
(11.3.16)
We see that the number of spatial dimensions cancels and that the entropy
bound is satisfied so long as
w ≤ 1.
(11.3.17)
This is an interesting result. Recall that the speed of sound within a
medium is given by
vs2 =
∂P
∂u
=w
Therefore, in the future, the bound will always be satisfied, since the speed
of sound is always less than the speed of light. The relation satisfied by
the scale factor vs2 = w ≤ 1 is just the usual causality requirement. As one
moves forward in time, the entropy bound then becomes more satisfied,
not less.
Next, go back in time using the black body radiation background as
the dominant entropy. Using a decoupling time tdecoupling ∼ 105 years
(when the background radiation fell out of equilibrium with the matter)
and extrapolating back using the previously calculated entropy relative to
the bound, one gets
1
2
S
−28 tdecoupling
= 10
(11.3.18)
A/(4G)
t
The entropy bound S =
tdecoupling
t
1
2
A
4G
is reached when
= 1028 ⇒ t =
tdecoupling
∼ 10−44 sec
1056
(11.3.19)
This time is comparable to the Planck time (by coincidence??). Therefore
the entropy bound is not exceeded after the Planck time.
11.4
Bousso’s Generalization
Consider an arbitrary cosmology. Take a space-like region Γ bounded
by the space-like boundary ∂Γ. At any point on the boundary we can construct four light rays that are perpendicular to the boundary[6]. We will
Entropy Bounds
115
call these the four branches. Two branches go toward the future. One of
them is composed of outgoing rays and the other is ingoing. Similarly two
branches go to the past. On any of these branches a light ray, together with
its neighbors define a positive or negative expansion as we move away from
the boundary. In ordinary flat space-time, if ∂Γ is convex the outgoing
(ingoing) rays have positive (negative) expansion. However in non-static
universes other combinations are possible. For example in a rapidly contracting universe both future branches have negative expansion while the
past branches have positive expansion.
If we consider general boundaries the sign of the expansion of a given
branch may vary as we move over the surface. For simplicity we restrict
attention to those regions for which a given branch has a unique sign. We
can now state Bousso’s rule: From the boundary ∂Γ construct all light
sheets which have negative expansion as we move away. These light sheets
may terminate at the tip of a cone or a caustic or even a boundary of the
geometry. Bousso’s bound states that the entropy passing through these light
sheets is less than A/4G where A is the boundary of ∂Γ.
To help visualize how Bousso’s construction works we will consider
spherically symmetric geometries and use Penrose diagrams to describe
them. The Penrose diagram represents the radial and time directions.
Each point of such a diagram really stands for a 2-sphere (more generally a (d − 1)-sphere). The four branches at a given point on the Penrose
diagram are represented by a pair of 45 degree lines passing through that
point. However we are only interested in the branches of negative expansion. For example in Figure 11.11 we illustrate flat space-time and the
negative expansion branches of a typical local 2-sphere. In general as we
move around in the Penrose diagram the particular branches which have
negative expansion may change. For example if the cosmology initially expands and then collapses, the outgoing future branch will go from positive
to negative expansion. Bousso introduced a notation to indicate this. The
Penrose diagram is divided into a number of regions depending on which
branches have negative expansion. In each region the negative expansion
branches are indicated by their directions at a typical point. Thus in Figure
11.12 we draw the Penrose–Bousso (PB) diagram for a positive curvature,
matter dominated universe that begins with a bang and ends with a crunch.
It consists of four distinct regions.
In Region I of Figure 11.12 the expansion of the universe causes both
past branches to have negative expansion. Thus we draw light surfaces
into the past. These light surfaces terminate on the initial boundary of the
116
Black Holes, Information, and the String Theory Revolution
Fig. 11.11
Negative expansion branches of 2-sphere in flat space-time
geometry and are similar to the truncated cones that we discussed in the
flat F.R.W. case. The holographic bound in this case says that the entropy
passing through either backward light surface is bounded by the area of the
2-sphere at point p. Bousso’s rule tells us nothing in this case about the
entropy on space-like surfaces bounded by p.
Now move on to Region II. The relevant light sheets in this region begin on the 2-sphere q and both terminate at the spatial origin. These
are untruncated cones and the entropy on both of them is holographically
bounded. There is something new in this case. We find that the entropy is
bounded on a future light sheet. Now consider a space-like surface bounded
by q and extending to the spatial origin (shown in Figure 11.13). It is evident that any matter which passes through the space-like surface must
also pass through the future light sheet. By the second law of thermodynamics the entropy on the space-like surface can not exceed the entropy on
the future light sheet. Thus in this case the entropy in a space-like region
can be holographically bounded. Therefore, one condition for a space-like
bound is that the entropy is bounded by a corresponding future light sheet.
With this in mind we return to Region I. For Region I there is no future
bound, and therefore the entropy is not bounded on space-like regions with
boundary p.
In Region III the entropy bounds are both on future light sheets. Nevertheless there is no space-like bound. The reason is that not all matter
Entropy Bounds
117
t = tfinal
Future
directed
Decreasing
area
CONTRACTION
III
Forward
light cone
TOWARDS
r = 0
Backwards
lightcone
TOWARDS
LARGE
RADIUS
q Point
corresponds
to finite area
of sphere
IV
Backwards
light cone
II
I
EXPANSION
p
t = 0
Fig. 11.12
Light cone
intercepts
singularity
Penrose–Bousso diagram for matter dominated universe
which pass through space-like surfaces are forced to pass through the future
light sheets.
Region IV is identical to Region II with the spatial origin being replaced
by the diametrically opposed antipode. Thus we see that there are light-like
bounds in all four regions but only in II and IV are there holographic bounds
on space-like regions. (See Figure 11.13.) Figure 11.14 demonstrates a
region that does not satisfy an entropy bound for this cosmology.
118
Black Holes, Information, and the String Theory Revolution
Entropy on this space
like surface is also
bounded by flux
through light-like
surface
Fig. 11.13
q
Bounding surfaces for inflowing entropy
t final
Area is INCREASING
in part of this light-like
surface. The entropy
thru such a surface is
NOT bounded in a
Robertson–Walker
cosmology
t = 0
Fig. 11.14
Light-like surface which does not satisfy entropy bound
Entropy Bounds
11.5
119
de Sitter Cosmology
de Sitter space holds special interest because of its close connection
with observational cosmology. The model of cosmology which is presently
gaining the status of a “standard model” involves de Sitter space in two
ways. First it is believed that the early universe underwent an epoch of
rapid inflation during which the space-time geometry was very close to de
Sitter. The inflationary theory is more than two decades old and by now
has been well tested.
More recently de Sitter space has entered cosmology as the likely candidate for the final fate of the universe. The history of the universe seems to
be a transition from an early de Sitter epoch in which the vacuum energy or
cosmological constant was very large to a late de Sitter phase characterized
by a very small but not zero cosmological constant.
de Sitter space is the solution of Einstein’s field equations with a positive
cosmological constant that exhibits maximal symmetry. Four-dimensional
de Sitter space may be defined by embedding it in (4 + 1) dimensional flat
Minkowski space. It is the hyperboloid given by
4
(xi )2 − (x0 )2 = R2 .
(11.5.20)
i=1
The radius of curvature R is related to the cosmological constant, λ.
R2 =
3
Gλ
(11.5.21)
de Sitter space can also be written in the form
dτ 2 = dt2 − a(t)2 dΩ23
(11.5.22)
where dΩ23 is the metric of a unit 3-sphere, and the scale factor a is given
by
a(t) = R cosh(t/R).
(11.5.23)
For our purposes we want to put this in a form that will allow us to
read off the Penrose diagram. To that end define the conformal time T by
dT = dt/a(t).
(11.5.24)
One easily finds that the geometry has the form
2
dτ 2 = (a(t)) (dT 2 − dΩ23 ).
(11.5.25)
120
Black Holes, Information, and the String Theory Revolution
0
−π
Fig. 11.15
π
0
Penrose diagram for de Sitter space-time
S
Fig. 11.16
Extremal space-like surface in de Sitter space-time
Furthermore when t varies between ±∞, the conformal time (T =
tan−1 (tanh(t/R)) − π2 ) varies from −π to 0. Since the polar angle on
the sphere also varies from 0 to π the Penrose diagram is a square as in
Figure 11.15.
Once again the Bousso construction divides the Penrose diagram into
four quadrants. However, the contracting light sheets in the upper and
lower quadrants are oriented oppositely to the usual F.R.W. case. The
reason is that the geometry rapidly expands as we move toward the upper
and lower boundary of the de Sitter space.
Of particular interest is the bound on a space-like surface beginning
at the center of the Penrose diagram and extending to the left edge as
in Figure 11.16. We leave it as an exercise to show that the entropy is
Entropy Bounds
121
t
Post-inflation
particle horizon
Larger area
Period of inflation
Fig. 11.17
Inflationary universe
bounded by the area of the 2-sphere at the center of the diagram. In fact
the maximum entropy on any such surface is given by
S=
4πR2
.
4G
(11.5.26)
de Sitter space has a special importance because of its role during the
early evolution of the universe. According to the inflationary hypothesis,
the universe began with a large vacuum energy which mimicked the effects
of a positive cosmological constant. During that period the geometry was de
Sitter space. But then at some time the vacuum energy began to decrease
and the universe made a transition to an F.R.W. universe. The transition
was accompanied by the production of a large amount of entropy, and is
called reheating.
Inflationary cosmology is illustrated in Figure 11.17. In order to construct the Penrose–Bousso diagram we begin by drawing Penrose diagrams
for both de Sitter space and either radiation or matter dominated F.R.W.
The Penrose–Bousso diagram for de Sitter space is shown in Figure 11.18.
In order to describe inflationary cosmology we must terminate the de Sitter
space at some late time and attach it to a conventional F.R.W. space as in
Figure 11.19. The dotted line where the two geometries are joined is the
reheating surface where the entropy of the universe is created.
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Black Holes, Information, and the String Theory Revolution
Point corresponds to
finite spherical area
Fig. 11.18
Buosso wedges for de Sitter geometry
Robertson–Walker
Space
de Sitter
Space
+
Join
t inflation
EXPANDING
Fig. 11.19
Reheating: Entropy is created
during reheating. There is no
entropy prior.
Joining of inflationary and post-inflationary geometries
Let us focus on the point p in Figure 11.20. It is easy to see that in
an ordinary inflationary cosmology p can be chosen so that the entropy on
the space-like surface p − q is bigger than the area of p. However Bousso’s
rule applied to point p only bounds the entropy on the past light sheet.
Entropy Bounds
Entropy flux
only after
reheating
123
Robertson–
Walker
p
q
REHEATING
No entropy
flux here
Inflation
Backwards
light cone
Fig. 11.20
Entropy bound on spherical region causally connected to inflationary
period
In this case most of the newly formed entropy on the reheating surface is
not counted since it never passed through the past light sheet. Typical
inflationary cosmologies can be studied to see that the past light sheet
bound is not violated.
11.6
Anti de Sitter Space
de Sitter space is important because it may describe the early and late
time behavior of the real universe. Anti de Sitter space is important for
an entirely different reason. It is the background in which the holographic
principle is best understood. In Chapter 12 the geometry of AdS will be
reviewed, but for our present purpose all we need are the properties of its
PB diagram.
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Black Holes, Information, and the String Theory Revolution
L
S
p
Boundary
r=0
ANY spacelike surface S will have
its entropy bounded by the area
Fig. 11.21
Static surface of large area in AdS space
AdS space is the vacuum of theories with a negative cosmological constant. The space-time in appropriate coordinates is static but unlike de
Sitter space the static coordinates cover the entire space. The vacuum of
AdS is a genuine zero temperature state.
The PB diagram consists of an infinite strip bounded on the left by the
spatial origin and on the right by a boundary. The PB diagram consists of a
single region in which both negative expansion light sheets point toward the
origin. Let us consider a static surface of large area A far from the spatial
origin. The surface is denoted by the dotted vertical line L in Figure 11.21.
We will think of L as an infrared cutoff. Consider an arbitrary point p on L.
Evidently Bousso’s rules bound the entropy on past and future light sheets
bounded by p. Therefore the entropy on any space-like surface bounded by
p and including the origin is also holographically bounded. In other words
the entire region to the left of L can be foliated with space-like surfaces
such that the maximum entropy on each surface is A/4G.
AdS space is an example of a special class of geometries which have
time-like Killing vectors and which can be foliated by space-like surfaces
that satisfy the Holographic bound. These two properties imply a very far
Entropy Bounds
125
reaching conclusion. All physics taking place in such backgrounds (in the
interior of the infrared cutoff L) must be described in terms of a Hamiltonian
that acts in a Hilbert space of dimensionality
Nstates = exp(A/4G)
(11.6.27)
The holographic description of AdS space is the subject of the next chapter.
126
Black Holes, Information, and the String Theory Revolution
Chapter 12
The Holographic Principle and Anti
de Sitter Space
12.1
The Holographic Principle
As we have seen in Chapter 11, the number of possible quantum states
in a region of flat space is bounded by the exponential of the area of the
region in Planck units. That fact together with the Ultraviolet/Infrared
connection and Black Hole Complementarity has led physics to an entirely
new paradigm about the nature of space, time and locality. One of the
elements of this paradigm is the Holographic Principle and its embodiment
in AdS space.
Let us consider a region of flat space Γ. We have seen that the maximum
entropy of all physical systems that can fit in Γ is proportional to the area
of the boundary ∂Γ, measured in Planck units. Typically, as in the case
of a lattice of spins, the maximum entropy is a measure of the number of
simple degrees of freedom∗ that it takes to completely describe the region.
This is almost always proportional to the volume of Γ. The exception
is gravitational systems. The entropy bound tells us that the maximum
number of non-redundant degrees of freedom is proportional to the area.
For a large macroscopic region this is an enormous reduction in the required
degrees of freedom. In fact if the linear dimensions of the system is of
order L then the number of degrees of freedom per unit volume scales like
1/L in Planck units. By making L large enough we can make the degrees
of freedom arbitrarily sparse in space. Nevertheless we must be able to
describe microscopic processes taking place anywhere in the region. One
way to think of this is to imagine the degrees of freedom of Γ as living
on ∂Γ with an area density of no more than ∼ 1 degree of freedom per
∗ By
a simple degree of freedom we mean something like a spin or the presence or absence
of a fermion. A simple degree of freedom represents a single bit of information.
127
128
Black Holes, Information, and the String Theory Revolution
Planck area. The analogy with a hologram is obvious; three-dimensional
space described by a two-dimensional hologram at its boundary! That this
is possible is called the Holographic Principle.
What we would ideally like to do is to have a solution of Einstein’s
equations that describes a ball of space with a spherical boundary and
then to count the number of degrees of freedom. Even better would be
to construct a description of the region in terms of a boundary theory
with a limited density of degrees of freedom. Ordinarily, it does not make
sense to consider a ball-like region with a boundary in the general theory
of relativity. But there is one special situation which is naturally ball-like.
It occurs when there is a negative cosmological constant; Anti de Sitter
space. Thus AdS is a natural framework in which to study the Holographic
Principle.
12.2
AdS Space
We saw in Chapter 11 that AdS space enjoys certain properties which
make it a natural candidate for a holographic Hamiltonian description. In
this lecture we will describe a very precise version of AdS “Holography”
which grew out of the mathematics of string theory. The remarkable precision is due to the unusually high degree of symmetry of the theory which
includes a powerful version of supersymmetry. However we will downplay
the mathematical aspects of the theory and concentrate on those physical
principles which are likely to be general.
The particular space that we will be interested in is not simple 5dimensional AdS but rather AdS(5)⊗S(5)[7][8][9]. This is a 10-dimensional
product space consisting of two factors, the 5-dimensional AdS and a
5-sphere S(5). Why the S(5)? The reason involves the high degree of
supersymmetry enjoyed by superstring theory. Generally the kind of supergravity theories that emerge from string theory don’t have cosmological
constants. But by bending some of the directions of space into compact
manifolds it becomes possible to generate a cosmological constant for the
resulting lower dimensional Kaluza–Klein type theory. From a conceptual
point of view the extra internal 5-sphere is not important. From a mathematical point of view it is essential if we want to be able to make precision
statements.
We will begin with a brief review of AdS geometry. For our purposes
5-dimensional AdS space may be considered to be a solid 4-dimensional
The Holographic Principle and Anti de Sitter Space
129
spatial ball times the infinite time axis. The geometry can be described by
dimensionless coordinates t, r, Ω where t is time, r is the radial coordinate
(0 ≤ r < 1) and Ω parameterizes the unit 3-sphere. The geometry has
uniform curvature R−2 where R is the radius of curvature. The metric we
will use is
dτ 2 =
'
(
R2
(1 + r2 )2 dt2 − 4dr2 − 4r2 dΩ2
(1 − r2 )2
(12.2.1)
There is another form of the metric which is in common use,
dτ 2 =
(
R2 ' 2
dt − dxi dxi − dy 2
2
y
(12.2.2)
where i runs from 1 to 3.
The metric in equation 12.2.2 is related to 12.2.1 in two different ways.
First of all it is an approximation to equation 12.2.1 in the vicinity of a
point on the boundary at r = 1. The 3-sphere is replaced by the flat
tangent plane parameterized by xi and the radial coordinate is replaced by
y, with y = (1 − r).
The second way that equations 12.2.1 and 12.2.2 are related is that
12.2.2 is the exact metric of an incomplete patch of AdS space. A time-like
geodesic can get to y = ∞ in a finite proper time so that the space in
equation 12.2.2 is not geodesically complete. It has a horizon at y = ∞.
When interpreted in this manner, time coordinates appearing in 12.2.1 and
12.2.2 are not the same.
The metric 12.2.2 may be expressed in terms of the coordinate z = 1/y.
)
*
1
(12.2.3)
dτ 2 = R2 z 2 (dt2 − dxi dxi ) − 2 dz 2
z
In this form it is clear that there is a horizon at z = 0 since the time–time
component of the metric vanishes there. The boundary is at z = ∞.
To construct the space AdS(5) ⊗ S(5) all we have to do is define 5 more
coordinates ω5 describing the unit 5-sphere and add a term to the metric
ds25 = R2 dω52
(12.2.4)
Although the boundary of AdS is an infinite proper distance from any
point in the interior of the ball, light can travel to the boundary and back
in a finite time. For example, it takes a total amount of (dimensionless)
time t = π for light to make a round trip from the origin at r = 0 to the
boundary at r = 1 and back. For all practical purposes AdS space behaves
130
Black Holes, Information, and the String Theory Revolution
like a finite cavity with reflecting walls. The size of the cavity is of order
R. In what follows we will think of the cavity size R as being much larger
than any microscopic scale such as the Planck or string scale.
Supplement on Properties of AdS metric
1) The point r = 0 is the center of the Anti de Sitter space and r varies from
2
0 to 1 in the space. A radial null geodesic satisfies 1 + r2 dt2 = 4dr2 ,
which means that a light beam will traverse the infinite proper distance
in r from 0 to 1 back to 0 in a round trip time given by π, which makes
the space causally finite.
2) The metric is singular at r = 1 in all components. A unit coordinate
time interval corresponds to increasingly large proper time intervals.
3) Near r = 1, the metric is approximately conformal, which means that
light rays move at 45◦ angles near the boundary.
ds2 ∼
=
4R2
(1−r 2 )2
'
(
−dt2 + dr2 + r2 dΩ2D−2
Light rays move slower (by a factor of 2) near the center of the Anti de
Sitter space.
4) Generally, the spatial metric is that of a uniformly (negatively) curved
space, a hyperbolic plane (or the Poincaré disk ).
12.3
Holography in AdS Space
We will refer to AdS(5) ⊗ S(5) as the bulk space and the 4-dimensional
boundary of AdS at y = 0 as the boundary. According to the Holographic
Principle we should be able to describe everything in the bulk by a theory
whose degrees of freedom can be identified with the boundary at y = 0.
However the Holographic Principle requires more than that. It requires
that the boundary theory has no more than 1 degree of freedom per Planck
area. To see what this entails, let us compute the area of the boundary.
From equation 12.2.1 we see that the metric diverges near the boundary.
The Holographic Principle and Anti de Sitter Space
131
Later we will regulate this divergence by moving in a little way from y = 0,
but for the time being we can assume that the number of degrees of freedom
per unit coordinate area is infinite. That suggests that the boundary theory
might be a quantum field theory, and that is in fact the case.
Another important fact involves the symmetry of AdS. Let us consider
the metric in the form 12.2.2. It is obvious that the geometry is invariant
under ordinary Poincaré transformations of the 4-dimensional Minkowski
coordinates t, xi . In addition there is a “dilatation” symmetry
t → λt
xi → λxi
y → λy
(12.3.5)
On the other hand if we consider the representation of AdS in 12.2.1 we
can see additional symmetry. For example the rotations of the sphere Ω are
symmetries. The full symmetry group of AdS(5) is the group O(4|2). In
addition there is also the symmetry O(6) associated with rotations of the
internal 5-sphere.
Since our goal is a holographic boundary description of the physics in
the bulk spacetime it is very relevant to ask how the symmetries act on
the boundary of AdS. Obviously, the 4-dimensional Poincaré symmetry
acts on the boundary straightforwardly. The dilatation symmetry also acts
as a simple dilatation of the coordinates t, x. All of the transformations
act on the boundary as conformal transformations which preserve light-like
directions on the boundary. In fact the full AdS symmetry group, when
acting on the boundary at y = 0 is precisely the conformal group of 4dimensional Minkowski space.
The implication of this symmetry of the boundary is that the holographic boundary theory must be invariant under the conformal group.
This together with the fact that the boundary has an infinite (coordinate)
density of degrees of freedom suggests that the holographic theory is a
Conformal Quantum Field Theory, and so it is.
As we mentioned, AdS(5) ⊗ S(5) is a solution of the 10-dimensional
supergravity that describes low energy superstring theory. Indeed the space
has more symmetry than just the conformal group and the O(6) symmetry
of the internal 5-sphere. The additional symmetry is the so-called N = 4
supersymmetry. This symmetry must also be realized by the holographic
theory. All of this leads us to the remarkable conclusion that quantum
gravity in AdS(5) ⊗ S(5) should be exactly described by an appropriate
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Black Holes, Information, and the String Theory Revolution
superconformal Lorentz invariant quantum field theory associated with the
AdS boundary.
In order to have a benchmark for the counting of degrees of freedom in
AdS(5) ⊗ S(5) imagine constructing a cutoff field theory in the bulk. A
conventional cutoff would involve a microscopic length scale such as the 10dimensional Planck length lp . One way to do this would be to introduce a
spatial lattice in 9-dimensional space. It is not generally possible to make a
regular lattice, but a random lattice with an average spacing lp is possible.
We can then define a simple theory such as a Hamiltonian lattice theory
on the space. In order to count degrees of freedom we also need to regulate
the area of the boundary of AdS which is infinite. To do so we introduce
a surface L at r = 1 − δ. The total 9-dimensional spatial volume in the
interior of L is easily computed using the metric 12.2.2, and is seen to be
critically divergent.
V (δ) ∼
R9
.
δ3
(12.3.6)
The number of bulk lattice sites and therefore the number of degrees of
freedom is
1 R9
V
∼ 3 9
9
lp
δ lp
(12.3.7)
In such a theory we also will find that the maximum entropy is of the same
order of magnitude. However the Holographic Principle suggests that this
entropy is overestimated.
The holographic bound discussed in Chapter 11 requires the maximum
entropy and the number of degrees of freedom to be of order
Smax ∼
A
lp8
(12.3.8)
where A is the 8-dimensional area of the boundary L. This is also easily
computed. We find
Smax ∼
1 R8
δ 3 lp8
(12.3.9)
In other words when R/lp becomes large the holographic description requires a reduction in the number of independent degrees of freedom by a
factor lp /R. To say it slightly differently, the Holographic Principle implies a complete description of all physics in the bulk of a very large AdS
The Holographic Principle and Anti de Sitter Space
133
space in terms of only lp /R degrees of freedom per spatial Planck volume.
Nonetheless the theory must be able to describe microscopic events in the
bulk even when R becomes extremely large.
12.4
The AdS/CFT Correspondence
The search for a holographic description of AdS(5) ⊗ S(5) is considerably narrowed by the symmetries. In fact there is only one known class of
systems with the appropriate N = 4 supersymmetry; the SU (N ) Supersymmetric Yang–Mills (SYM) theories.
The correspondence between gravity or its string theoretic generalization in AdS(5) ⊗ S(5) and Super Yang–Mills (SYM) theory on the
boundary is the subject of a vast literature. We will only review some
of the salient features. The correspondence states that there is a complete
equivalence between superstring theory in the bulk of AdS(5) ⊗ S(5) and
N = 4, 3 + 1-dimensional, SU (N ), SYM theory on the boundary of the
AdS space[7][8][9]. In these lectures SYM theory will always refer to this
particular version of supersymmetric gauge theory, N represents the number of supersymmetries, and N is the dimension of the Yang–Mills gauge
theory.
It is well known that SYM is conformally invariant and therefore has no
dimensional parameters. It will be convenient to define the theory to live
on the boundary parametrized by the dimensionless coordinates t, Ω or t, x.
The corresponding momenta are also dimensionless. In fact we will use the
convention that all SYM quantities are dimensionless. On the other hand
the bulk gravity theory quantities such as mass, length and temperature
carry their usual dimensions. To convert from SYM to bulk variables, the
conversion factor is R. Thus if ESYM and M represent the energy in the
SYM and bulk theories
ESYM = M R.
Similarly bulk time intervals are given by multiplying the t interval by R.
There is one question that may be puzzling to the reader. Since
AdS(5) ⊗ S(5) is a 10-dimensional spacetime one might think that its
boundary is (8 + 1) dimensional. But there is an important sense in which
it is 3 + 1 dimensional. To see this let us Weyl rescale the metric by a
134
Black Holes, Information, and the String Theory Revolution
2
R
factor (1−r
2 )2 so that the rescaled metric at the boundary is finite. The
new metric is
( '
(
'
(12.4.10)
dS 2 = (1 + r2 )2 dt2 − 4dr2 − 4r2 dΩ2 + (1 − r2 )2 dω52
Notice that the size of the 5-sphere shrinks to zero as the boundary at r = 1
is approached. The boundary of the geometry is therefore 3+1 dimensional.
Let us return to the correspondence between the bulk and boundary theories. The 10-dimensional bulk theory has two dimensionless parameters.
These are:
1) The radius of curvature of the AdS space measured in string units R/ s .
Alternately we could measure R in 10-dimensional Planck units. The
relation between string and Planck lengths is given by
g2
8
s
= lp8
2) The dimensionless string coupling constant g.
The string coupling constant and length scale are related to the 10dimensional Planck length and Newton constant by
lp8 = g 2
8
s
=G
(12.4.11)
On the other side of the correspondence, the gauge theory also has two
constants. They are
1) The rank of the gauge group N
2) The gauge coupling gym
Obviously the two bulk parameters R and g must be determined by N
and gym . In these lectures we will assume the relation that was originally
derived in [7].
R
s
1
2
= (N gym
)4
2
g = gym
(12.4.12)
We can also write the 10-dimensional Newton constant in the form
G = R8 /N 2
(12.4.13)
There are two distinct limits that are especially interesting, depending on one’s motivation. The AdS/CFT correspondence has been widely
studied as a tool for learning about the behavior of gauge theories in the
The Holographic Principle and Anti de Sitter Space
135
strongly coupled ’t Hooft limit. From the gauge theory point of view the
’t Hooft limit is defined by
gym → 0
N →∞
2
N = constant
gym
(12.4.14)
From the bulk string point of view the limit is
g→0
R
= constant
(12.4.15)
s
Thus the strongly coupled ’t Hooft limit is also the classical string theory
limit in a fixed and large AdS space. This limit is dominated by classical
supergravity theory.
The interesting limit from the viewpoint of the holographic principle is
a different one. We will be interested in the behavior of the theory as the
AdS radius increases but with the parameters that govern the microscopic
physics in the bulk kept fixed. This means we want the limit
g = constant
s → ∞
(12.4.16)
gym = constant
N →∞
(12.4.17)
R/
On the gauge theory side this is
Our goal will be to show that the number of quantum degrees of freedom in
the gauge theory description satisfies the holographic behavior in equation
12.3.8.
12.5
The Infrared Ultraviolet Connection
In either of the metrics in equation 12.2.1 or 12.2.2 the proper area of any
finite coordinate patch tends to ∞ as the boundary of AdS is approached.
Thus we expect that the number of degrees of freedom associated with
such a patch should diverge. This is consistent with the fact that a continuum quantum field theory such as SYM has an infinity of modes in any
finite three-dimensional patch. In order to do a more refined counting[9]
136
Black Holes, Information, and the String Theory Revolution
we need to regulate both the area of the AdS boundary and the number
of ultraviolet degrees of freedom in the SYM. As we will see, these apparently different regulators are really two sides of the same coin. We have
already discussed infrared (IR) regulating the area of AdS by introducing
a surrogate boundary L at r = 1 − δ or similarly at y = δ.
That the IR regulator of the bulk theory is equivalent to an ultraviolet
(UV) regulator in the SYM theory is called the IR/UV connection[9]. It
is in many ways similar to the behavior of strings as we study them at
progressively shorter time scales. In Chapter 14 we will find the interesting
behavior that a string appears to grow as we average its properties over
smaller and smaller time scales. To understand the relation between this
phenomenon and the IR/UV connection in AdS we need to discuss the
relation between AdS(5) ⊗ S(5) and D-branes.
D-branes are objects which occur in superstring theory. They are stable “impurities” of various dimensionality that can appear in the vacuum.
A Dp-brane is a p-dimensional object. We are especially interested in
D3-branes. Such objects fill 3 dimensions of space and also time. Their
properties are widely studied in string theory and we will only quote the
results that we need. The most important property of D3-branes is that
they are embedded in a 10-dimensional space. Let us assume that they fill
time and the 3 spatial coordinates xi . Let the other 6 coordinates be called
√
z m and let z ≡ z m z m . We will place a “stack” of N D3-branes at z = 0.
Now a single D-brane has local degrees of freedom. For example the
location in z may fluctuate. Thus we can think of the z location as a
scalar field living on the D-brane. In addition there are modes of the brane
which are described by vector fields with components in the t, x direction as
well as fermionic modes needed for supersymmetry. Our main concern will
be with the z(x, t) fluctuations whose action is known from string theory
calculations to be a that of conventional 3+1 dimensional scalar field theory.
D-branes can also be juxtaposed to form stacks of D-branes. A stack of
N D-branes has a mass and D-brane charge which grow with N . The mass
and charge are sources of bulk fields such as the gravitational field. What
makes the D-brane stack interesting to us is that the geometry sourced by
the stack is exactly that of AdS(5) ⊗ S(5). In fact the geometry defined by
12.2.3 and 12.2.4 is closely related to that of a D-brane stack.
Specifically the geometry sourced by the D-branes is a particular solution of the supergravity equations of motion:
ds2 = F (z)(dt2 − dx2 ) − F (z)−1 dzdz
(12.5.18)
The Holographic Principle and Anti de Sitter Space
137
where
−1/2
cgs N
F (z) = 1 +
z4
and c is a numerical constant. If we consider the limit in which
then we can replace F (z) by the simpler expression
F (z) ∼
=
z2
1
(cgs N ) 2
.
(12.5.19)
cgs N
z4
>> 1
(12.5.20)
It is then a simple exercise to see that the D-brane metric is of the form
12.2.3, 12.2.4.
Furthermore the theory of the fluctuations of the stack is N = 4 SYM.
All of the fields in this theory form a single supermultiplet belonging to the
adjoint (N × N matrix) representation of SU (N ).
In this lecture we give an argument for the IR/UV connection based
on the quantum fluctuations of the positions of the D3-branes which are
nominally located at the origin of the coordinate z in equation 12.2.3. The
location of a point on a 3-brane is defined by six coordinates z, ω5 . We may
also choose the six coordinates to be Cartesian coordinates (z 1 , ..., z 6 ). The
original coordinate z is defined by
z 2 = (z 1 )2 + ... + (z 6 )2
(12.5.21)
As we indicated, the coordinates z m are represented in the SYM theory
by six scalar fields on the world volume of the branes. If the six scalar fields
φn are canonically normalized, then the precise connection between the z s
and φ s is
z=
gym 2s
φ
R2
(12.5.22)
Strictly speaking equation 12.5.22 does not make sense because the fields
φ in SU (N ) are N × N matrices, where we identify the N eigenvalues of
the matrices in equation 12.5.18 to be the coordinates z m of the N D3branes[10]. The geometry is noncommutative and only configurations in
which the six matrix valued fields commute have a classical interpretation.
√
However the radial coordinate z = z m z m can be defined by
z2 =
gym 2s
R2
2
1
T rφ2
N
(12.5.23)
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Black Holes, Information, and the String Theory Revolution
A question which is often asked is: where are the D3-branes located in
the AdS space? The usual answer is that they are at the horizon z = 0.
However our experiences in Chapter 14 with similar questions will warn us
that the answer may be more subtle. What we will find there is that the
way information is localized in space depends on what frequency range it
is probed with. High frequency or short time probes see the string widely
spread in space while low frequency probes see a well localized string.
To answer the corresponding question about D3-branes we need to study
the quantum fluctuations of their position. The fields φ are scalar quantum
fields whose scaling dimensions are known to be exactly (length)−1 . From
this it follows that any of the N 2 components of φ satisfies
φ2ab ∼ δ −2
(12.5.24)
where δ is the ultraviolet regulator of the field theory. It follows from
equation 12.5.20 that the average value of z satisfies
2
gym 2s
N
< z >2 ∼
(12.5.25)
R2
δ2
or, using equation 12.4.12
< z >2 ∼ δ −2
(12.5.26)
In terms of the coordinate y which vanishes at the boundary of AdS
< y >2 ∼ δ 2 .
(12.5.27)
Here it is seen that the location of the brane is given by the ultraviolet
cutoff of the field theory on the boundary. Evidently low frequency probes
see the branes at z = 0 but as the frequency of the probe increases the brane
appears to move toward the boundary at z = ∞. The precise connection
between the UV SYM cutoff and the bulk theory IR cutoff is given by
equation 12.5.23.
12.6
Counting Degrees of Freedom
Let us now turn to the problem of counting the number of degrees of
freedom needed to describe the region y > δ [9]. The UV/IR connection
implies that this region can be described in terms of an ultraviolet regulated
theory with a cutoff length δ. Consider a patch of the boundary with
The Holographic Principle and Anti de Sitter Space
139
unit coordinate area. Within that patch there are 1/δ 3 cutoff cells of size
δ. Within each such cell the fields are constant in a cutoff theory. Thus
each cell has of order N 2 degrees of freedom corresponding to the N ⊗ N
components of the adjoint representation of U (N ). Thus the number of
degrees of freedom on the unit area is
N2
(12.6.28)
δ3
On the other hand the 8-dimensional area of the regulated patch is
Ndof ≈
R3
R8
5
×
R
=
δ3
δ3
and the number of degrees of freedom per unit area is
A=
Ndof
N2
∼ 8
A
R
Finally we may use equation 12.4.13
(12.6.29)
(12.6.30)
Ndof
1
∼
(12.6.31)
A
G
This is very gratifying because it is exactly what is required by the Holographic Principle.
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Black Holes, Information, and the String Theory Revolution
Chapter 13
Black Holes in a Box
The apparently irreconcilable demands of black hole thermodynamics
and the principles of quantum mechanics have led us to a very strange view
of the world as a hologram. Now we will return, full circle, to see how the
holographic description of AdS(5) ⊗ S(5) provides a description of black
holes.
We have treated Schwarzschild black holes as if they were states of
thermal equilibrium, but of course they are not. They are long-lived objects,
but eventually they evaporate. We can try to prevent their evaporation by
placing them in a thermal heat bath at their Hawking temperature but
that does not work. The reason is that their specific heat is negative;
their temperature decreases as their energy or mass increases. Any object
with this property is thermodynamically unstable. To see this, suppose a
fluctuation occurs in which the black hole absorbs an extra bit of energy
from the surrounding heat bath. For an ordinary system with positive
specific heat this will raise its temperature which in turn will cause it to
radiate back into the environment. The fluctuations are self-regulating.
But a system with negative specific heat will lower its temperature when
it absorbs energy and will become cooler than the bath. This in turn will
favor an additional flow of energy from the bath to the black hole and a
runaway will occur. The black hole will grow indefinitely. If on the other
hand the black hole gives up some energy to the environment it will become
hotter than the bath. Again a runaway will occur that leads the black hole
to disappear.
A well known way to stabilize the black hole is to put it in a box so that
the environmental heat bath is finite. When the black hole absorbs some
energy it cools but so does the finite heat bath. If the box is not too big
the heat bath will cool more than the black hole and the flow of heat will
141
142
Black Holes, Information, and the String Theory Revolution
be back to the bath. In this lecture we will consider the properties of black
holes which are stabilized by the natural box provided by Anti de Sitter
space. More specifically we consider large black holes in AdS(5)⊗S(5) and
their holographic description in terms of the N = 4 Yang–Mills theory.
The black holes which are stable have Schwarzschild radii as large or
larger than the radius of curvature R. They homogeneously fill the 5-sphere
and are solutions of the dimensionally reduced 5-dimensional Einstein equations with a negative cosmological constant. The thermodynamics can be
derived from the black hole solutions by first computing the area of the
horizon and then using the Bekenstein–Hawking formula.
Before writing the AdS–Schwarzschild metric, let us write the metric of
AdS in a form which is convenient for generalization.
dτ 2 =
−1
r2
r2
1 + 2 dt2 − 1 + 2
dr2 − r2 dΩ2
R
R
(13.0.1)
where in this formula r runs from 0 to the boundary at r = ∞. Note that
the coordinates r, t are not the same as in equation 12.2.1.
2
The AdS black hole is given by modifying the function 1 + Rr 2 :
dτ 2 =
−1
r2
µG
r2
µG
1 + 2 − 5 2 dt2 − 1 + 2 − 5 2
dr2 −r2 dΩ2 (13.0.2)
R
R r
R
R r
where the parameter µ is proportional to the mass of the black hole and G
is the 10-dimensional Newton constant. The horizon of the black hole is at
the largest root of
µG
r2
1+ 2 − 5 2 =0
R
R r
The Penrose diagram of the AdS black hole is shown in Figure 13.1. One
finds that the entropy is related to the mass by
1
S = c M 3 R11 G−1 4
(13.0.3)
where c is a numerical constant. Using the thermodynamic relation dM =
T dS we can compute the relation between mass and temperature:
M =c
R11 T 4
G
(13.0.4)
Black Holes in a Box
Fig. 13.1
143
Penrose diagram of the AdS black hole
or in terms of dimensionless SYM quantities
Esym = c
R8 4
T
G sym
4
= cN 2 Tsym
(13.0.5)
Equation 13.0.5 has a surprisingly simple interpretation. Recall that
in 3 + 1 dimensions the Stephan–Boltzmann law for the energy density of
radiation is
E ∼ T 4V
(13.0.6)
where V is the volume. In the present case the relevant volume is the
dimensionless 3-area of the unit boundary sphere. Furthermore there are
∼ N 2 quantum fields in the U (N ) gauge theory so that apart from a numerical constant equation 13.0.5 is nothing but the Stephan–Boltzmann
law for black body radiation. Evidently the holographic description of the
AdS black holes is as simple as it could be; a black body thermal gas of N 2
species of quanta propagating on the boundary hologram.
The constant c in equation 13.0.3 can be computed in two ways. The
first is from the black hole solution and the Bekenstein–Hawking formula.
The second way is to calculate it from the boundary quantum field theory in
the free field approximation. The calculations agree in order of magnitude,
but the free field gives too big a coefficient by a factor of 4/3. This is
not too surprising because the classical gravity approximation is only valid
when gY2 M N is large.
144
13.1
Black Holes, Information, and the String Theory Revolution
The Horizon
The high frequency quantum fluctuation of the location of the D3-branes
are invisible to a low frequency probe. Roughly speaking this is insured by
the renormalization group as applied to the SYM description of the branes.
The renormalization group is what insures that our bodies are not severely
damaged by constant exposure to high frequency vacuum fluctuations. We
are not protected in the same way from classical fluctuations. An example is the thermal fluctuations of fields at high temperature. All probes
sense thermal fluctuations of the brane locations. Let us return to equation
12.5.24 but now, instead of using equation 12.5.25 we use the thermal field
fluctuations of φ. For each of the N 2 components the thermal fluctuations
have the form
2
< φ2 >= Tsym
(13.1.7)
and we find equations 12.5.26 and 12.5.27 replaced by
2
< z >2 ∼ Tsym
2
−2
< y > ∼ Tsym
(13.1.8)
It is clear that the thermal fluctuations will be strongly felt out to a coordinate distance z = Tsym . In terms of r the corresponding position is
1 − r ∼ 1/Tsym
(13.1.9)
In fact this coincides with the location of the horizon of the AdS black hole.
13.2
Information and the AdS Black Hole
The AdS black hole is an ideal laboratory for investigating how bulk
quantum field theory fails when applied to the fine details of Hawking radiation. Let us consider some field that appears in the supergravity description of the bulk. Such objects are 10-dimensional fields and should
not be confused with the 4-dimensional quantum fields associated with the
boundary theory. A simple example is the minimally coupled scalar dilaton
field φ. We will only consider dilaton fields which are constant on the 5sphere. In that case the action for φ is the minimally coupled scalar action
Black Holes in a Box
145
in 5-dimensional AdS.
I=
√
d5 x −gg µν ∂µ φ∂ν φ.
(13.2.10)
The appropriate boundary conditions are φ → 0 at the boundary of the
AdS, i.e. r → ∞.
Plugging in the black hole metric given in equation 13.0.2, we find a
number of things. First φ(r) ∼ r−4 Φ as r → ∞. It is the value of Φ on
the boundary which is identified with a local field in the boundary Super
Yang–Mills theory. This is true both in pure AdS as well as in the AdS–
Schwarzschild metrics. Secondly, in the pure AdS background φ is periodic
in time, but in the black hole metric φ goes to zero exponentially with time:
φ → exp−γt.
(13.2.11)
Equation 13.2.11 has implications for quantum correlation functions in
the black hole background. Consider the correlator φ(t)φ(t ) . Equation
13.2.11 requires it to behave like
φ(t)φ(t ) ∼ exp−γ|t − t |.
(13.2.12)
for large |t − t | The parameter γ depends on the black hole mass or temperature and has the form
γ=
H(µGR−7 )
R
(13.2.13)
where H is a dimensionless increasing function of its argument.
The meaning of this exponential decrease of the correlation function is
that the effects of an initial perturbation at time t dissipate away and are
eventually lost. In other words the system does not preserve any memory
of the initial perturbation. This type of behavior is characteristic of large
thermal systems where γ would correspond to some dissipation coefficient.
However, exponential decay is not what is really expected for systems of
finite entropy such as the AdS black hole that we are dealing with. Any
quantum system with finite entropy preserves some memory of a perturbation. Since AdS is exactly described by a conventional quantum system it
follows that the correlator should not go to zero. We shall now prove this
assertion.
The essential point is that any quantum system with finite thermal
entropy must have a discrete spectrum. This is because the entropy is
essentially the logarithm of the number of states per unit energy. Indeed
146
Black Holes, Information, and the String Theory Revolution
the spectrum of the boundary quantum field theory is obviously discrete
since it is a theory defined on a finite sphere.
Let us now consider a general finite closed system described by a thermal
density matrix and a thermal correlator of the form
F (t) = A(0)A(t) =
1
T re−βH A(0)eiHt A(0)e−iHt .
Z
(13.2.14)
By finite we simply mean that the spectrum is discrete and the entropy
finite. Inserting a complete set of (discrete) energy eigenstates gives
F (t) =
1 −βEi i(Ej −Ei )t
e
e
|Aij |2 .
Z ij
(13.2.15)
For simplicity we will assume that the operator A has no matrix elements
connecting states of equal energy. This means that the time average of F
vanishes.
Let us now consider the long time average of F (t)F ∗ (t).
1
T →∞ 2T
+T
L = lim
dtF (t)F ∗ (t)
(13.2.16)
−T
Using equation 13.2.15 it is easy to show that the long time average is
L=
1 −β(Ei +Ek )
e
|Aij |2 |Akl |2 δ(Ej −El +Ek −Ei ) .
Z2
(13.2.17)
ijkl
where the delta function is defined to be zero if the argument is nonzero and
1 if it is zero. The long time average L is obviously nonzero and positive.
Thus it is not possible for the correlator F (t) to tend to zero as the time
tends to infinity and the limits required by the AdS/CFT correspondence
cannot exist. The value of the long time average for such finite systems can
be estimated, and it is typically of the order e−S where S is the entropy
of the system. This observation allows us to understand why it tends to
zero in the (bulk) QFT approximation. In studying QFT in the vicinity
of a horizon we have seen that the entropy is UV divergent. This is due
to the enormous number of short wave length modes near the horizon.
This leads us to a very important and general conclusion: any phenomenon
which crucially depends on the finiteness of horizon entropy will be gotten
wrong by the approximation of QFT in a fixed background. This includes
questions of information loss and of particular interest in this lecture, the
long time behavior of correlation functions.
Black Holes in a Box
147
How exactly do the correlations behave in the long time limit? The
answer is not that they uniformly approach constants given by the long
time averages. The expected behavior is that they fluctuate chaotically. A
large fluctuation which reduces the entropy by amount ∆S has probability
e−∆S . Thus we can expect large fluctuations in the correlators at intervals
of order eS . These fluctuations are analogous to the classical phenomenon
of Poincaré recurrences. It is generally found that the large time behavior
of correlators is chaotic “noise” with the long time average given by
L ∼ e−S .
(13.2.18)
This long time behavior, missed by bulk quantum field theory, is a small
part of the encoding of information in the thermal atmosphere of the AdS
black hole.
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Black Holes, Information, and the String Theory Revolution
PART 3
Black Holes and Strings
Chapter 14
Strings
We have learned a great deal about black holes by considering the behavior of quantum fields near horizons. But ultimately local quantum field
theory fails in a number of ways. In general the failures can be attributed to
a common cause – quantum field theory has too many degrees of freedom.
The earliest evidence that QFT is too rich in degrees of freedom was
the uncontrollable short distance divergences in gravitational perturbation
theory. As a quantum field theory, Einstein’s general relativity is very badly
behaved in the ultraviolet.
Even more relevant for our purposes is the divergence in the entropy
per unit area of horizons that was found in Chapter 4. Entropy is a direct
measure of the number of active degrees of freedom of a system. Evidently
there are far too many degrees of freedom very close to a horizon in QFT.
Later in Chapter 12 we quantified just how over-rich QFT is.
The remaining portion of this book deals, in an elementary way, with
a theory that seems to have just the right number of degrees of freedom:
string theory. The problems posed by black holes for a fundamental theory
of quantum gravity are non-perturbative. Until relatively recently, string
theory was mostly defined by a set of perturbation rules. Nevertheless, even
in perturbation theory, we will see certain trends that are more consistent
with black hole complementarity than the corresponding trends in QFT.
In Chapter 9 we explained that the key to understanding black hole
complementarity lies in the ultrahigh frequency oscillations of fluctuations
of matter in its own rest frame. The extreme red shift between the freely
falling frame and the Schwarzschild frame may take phenomena which are
of too high frequency to be visible ordinarily and make them visible to the
outside observer. As an example, imagine a freely falling whistle that emits
a sound of such high frequency that it cannot be heard by the human ear.
151
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Black Holes, Information, and the String Theory Revolution
As the whistle approaches the horizon, the observer outside the black hole
hears the frequency red shifted. Eventually it becomes audible, no matter
how high the frequency in the whistle’s rest frame.
On the other hand, the freely falling observer who accompanies the
whistle never gets the benefit of the increasing red shift. She never hears
the whistle.
This suggests that the consistency of black hole complementarity is a
deep constraint on how matter behaves at very short times or high frequencies. Quantum field theory gets it wrong, but string theory seems to do
better. The qualitative behavior of strings is the subject of this lecture.
In order to compare string theory and quantum field theory near a
horizon, we will first study the case of a free particle falling through a
Rindler horizon. As we will see, it is natural to use light cone coordinates
for this problem. The process and conventions are illustrated in Figure
14.1. The coordinates X ± are defined by
ρ
X0 ± X
√
= ∓ √ e∓τ
2
2
(14.0.1)
2
dτ 2 = 2dX + dX − − dX i
(14.0.2)
X± =
and the metric is given by
where X i run over the coordinates in the plane of the horizon. We will
refer to X i as the transverse coordinates, because they are transverse to
the direction of motion of the point particle. The trajectory of the particle
is taken to be
Xi = 0
X− − X+ =
√
2L
(14.0.3)
As the particle falls closer and closer to the horizon, the constant τ
surfaces become more and more light-like in the particle’s rest frame. In
other words, the particle and the Rindler observer are boosted relative to
one another by an ever-increasing boost angle.
Near the particle trajectory X + and τ are related by
X+ ∼
= −2L e−2τ
(14.0.4)
for large τ . This suggests that the description of mechanics in terms of the
Rindler (or Schwarzschild) time be replaced by a description in light cone
Strings
X
153
-
X
+
2
L
L
Fig. 14.1
Free particle falling through a Rindler horizon
coordinates with X + playing the role of the independent time coordinate.
We will therefore briefly review particle mechanics in the light cone frame.
14.1
Light Cone Quantum Mechanics
In order to write the action for a relativistic point particle we introduce
a parameter σ along the world line of the particle. Since the action only
depends on the world line and not the way we parameterize it, the action
should be invariant under a reparameterization. Toward this end we also
154
Black Holes, Information, and the String Theory Revolution
introduce an “einbein” e(σ) that transforms under σ-reparameterizations.
e1 (σ 1 ) dσ 1 = e(σ) dσ
(14.1.5)
The action is given by
W =
L =
L dσ
− 12
1 dX µ dXµ
e dσ dσ
− em
2
(14.1.6)
where m is the mass of the particle. The action in equation 14.1.6 is
invariant under σ-reparameterizations.
Let us now write equation 14.1.6 in terms of light cone coordinates and,
at the same time use our gauge freedom to fix σ = X + (which is then
treated as a time variable). Then L takes the form
1 dX i dX i
1
2 dX −
+
− e m2
L =
(14.1.7)
−
2
e dσ
e dσ dσ
The conserved canonical momenta are given by
P− =
∂L
∂ Ẋ −
= − 1e
(14.1.8)
Pi =
∂L
∂ Ẋ i
=
1
e
Ẋ i
where dot refers to σ derivative. Note that the conservation of P− insures
that e(σ) has a fixed constant value.
The Hamiltonian is easily obtained by the standard procedure:
H =
m2 e
ePi2
+
2
2
(14.1.9)
This form of H manifests a well known fact about light cone physics. If
we focus on the transverse degrees of freedom, the Hamiltonian has all the
properties of a non-relativistic system with Galilean symmetry. The second term in H is just a constant, and can be interpreted as an internal
energy that has no effect on the transverse motion. The first term has
the usual non-relativistic form with e−1 playing the role of an effective
transverse mass. This Hamiltonian and its associated quantum mechanics exactly describes the point particles of conventional free quantum field
theory formulated in the light cone gauge.
Now let us consider the transverse location of the particle as it falls
toward the horizon. In particular, suppose the particle is probed by an
Strings
155
experiment which takes place over a short time interval δ just before horizon
crossing. In other words, the particle is probed over the time interval
−δ < X + < 0
(14.1.10)
by a quantum of (Minkowski) energy ∼ 1δ . This experiment is similar to the
one discussed in Chapter 9.3, except that the probe carries out information
about the transverse location of the particle instead of its baryon number.
Since the interaction is spread over the time interval in equation 14.1.10,
the instantaneous transverse position should be replaced by the time averaged coordinate Xδi
Xδi =
1
δ
0
X i (σ) dσ
(14.1.11)
−δ
To evaluate equation 14.1.11, we use the non-relativistic equations of motion
X i (σ) = X i (0) + eP i σ
(14.1.12)
to give
Xδi = X i (0) +
eP i δ
.
2
(14.1.13)
Finally, let us suppose that the particle wave function is initially a
smooth wave packet well localized in transverse position with uncertainty
∆X i . Let us also assume the very high momentum components of the wave
function are negligible. Under these conditions nothing singular happens
to the probability distribution for Xδi as δ → 0. No matter how small δ
is, the effective probability distribution for Xδ is concentrated in a well
localized region of fixed extent, δX. There is no tendency for information
to transversely spread over a stretched horizon.
All of this is exactly what is expected for an ordinary particle in free
quantum field theory. For the more interesting case of an interacting quantum field theory, we could study the transverse properties of an interacting
or composite particle such as a hydrogen atom. For example, a time averaged relative coordinate or charge density can be defined, and it too shows
no sign of spreading as the sampling interval δ tends to zero.
Why is this a problem? The reason is that it conflicts with the complementarity principle. Complementarity requires the probe to report that
the particle fell into a very high temperature environment in which it repeatedly suffered high energy collisions. In this kind of environment the
information stored in the infalling system would be thermalized and spread
156
Black Holes, Information, and the String Theory Revolution
over the horizon. The implication for the probing experiment is that the
particle should somehow spread or diffuse roughly the way the effective
charge distribution did in Chapter 7.
14.2
Light Cone String Theory
Although naive pertubative string theory cannot capture this effect completely correctly, the tendency is already there in the theory of free strings.
A free string is a generalization of a free particle. There are a number of
excellent textbooks on string theory that the reader who is interested in
technical details can consult. For our purposes, only the most elementary
aspects of string theory will be needed.
A string is a one-dimensional continuum whose points are parameterized
by a continuous parameter σ 1 . The transverse coordinates of the point at
σ 1 are labeled X i (σ), where σ 1 runs from 0 to 2π. It is also a function of a
time-like parameter σ 0 , which is identified with light cone time X + . Thus
X i (σ 0 , σ 1 ) is a field defined on a 1+1 dimensional parameter space (σ a ).
In addition to X i (σ), the canonical momentum density Pi (σ) can also be
defined. At equal times X and P satisfy
i
(14.2.14)
X (σ), Pj (σ ) = i δji δ(σ − σ )
The light cone Hamiltonian for the free string is a natural generalization of
that for a free particle;
2 2π
∂X i
dσ 1
2
|Pi (σ )| +
(14.2.15)
H =
P− 0
2
∂σ We have used units in which the string tension (energy per unit length in
the rest frame) is unity.
The equation of motion following from equations 14.2.14 and 14.2.15 is
a simple wave equation
∂2X i
2
(∂σ 0 )
−
∂2X i
2
(∂σ 1 )
= 0.
(14.2.16)
Quantization of the string is straightforward. X i (σ) becomes a free scalar
field in 1+1 dimensions satisfying equation 14.2.16 with periodic boundary
conditions in σ 1 , X(σ 0 , 2π) = X(σ 0 , 0).
Strings
157
The string differs in important ways from the free particle, especially in
its short time behavior. As we have repeatedly emphasized, it is the short
time behavior that is key to complementarity.
Let us consider the analog of the question that we addressed about the
time averaged location of the point particle. Now we consider the time
averaged location of a point on the string. Thus, define
Xδ =
1
δ
δ
dσ 0 X(σ)
(14.2.17)
0
Since all points σ 1 are equivalent, it doesn’t matter what value σ 1 takes
on the right hand side when we evaluate Xδ . A useful measure of how
much the information in a string is spread as it falls towards the horizon is
provided by the fluctuations in Xδ , that is
+ ,
2
(14.2.18)
(∆X)2 = Xδ2 − Xδ
The state used for the expectation value in equation 14.2.18 is the ground
state string. This quantity is easily calculated and diverges logarithmically
as δ → 0. In other words, as the string approaches the horizon, any experiment (from the outside) to determine how its internal parts are distributed
will indicate a logarithmic increase in the area it occupies
(∆X)2 ∼ | log δ |.
(14.2.19)
Another way to write equation 14.2.19 is to use the connection between
Rindler time and light cone time in equation 14.0.4
(∆X)2 ∼ | log (2Le−2τ ) | ∼ 2τ.
Finally, we can use the relation betwen Rindler time and Schwarzschild
time given by τ = t/4M G to obtain
(∆X)2 ∼
α t
.
4M G
(14.2.20)
In equation 14.2.20 we have restored the units by including the factor α ,
the inverse string tension.
Here we see the beginnings of an explanation of complementarity. The
observer outside the black hole will find the string diffusing over an increasing area of the horizon as time progresses. But an observer falling with the
string and doing low energy experiments on it would conclude that the
string remains a fixed finite size as it falls.
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Black Holes, Information, and the String Theory Revolution
The linear growth of the area in equation 14.2.20 is much slower than the
growth of a charged particle described in Chapter 7. In that case inspection
of 7.0.21 indicates that the growth is exponential. A completely consistent
theory would require these growth patterns to match. The true exponential
asymptotic growth is undoubtedly a non-perturbative phenomenon that
involves string interactions in an essential way.
To see how interactions influence the evolution, let’s determine the average total length of string, projected onto the two-dimensional transverse
plane
2π
i 2
1 ∂X dσ 1 (14.2.21)
=
∂σ
0
i 2
As a preliminary, let us consider the ground state average of ∂X
∂σ1 . This
is another exercise in free scalar quantum field theory, and the result is
quadratically divergent.
i
over the time interval δ, we find that the
If however ∂X
∂σ is averaged
i 2
∂X ground state average of ∂σ1 is given by
-
∂X i ∂X i
∂σ 1 ∂σ 1
.
∼
1
δ2
(14.2.22)
i
∂X
Using the fact that the probability-distribution
. for ∂σ1 is Gaussian in free
∂X1 2 or scales as
field theory, we can conclude that
∂σ
1
δ
(14.2.23)
1 2τ
e .
2L
(14.2.24)
≈
or using equation 14.0.4
≈
In other words, as the string falls toward the horizon, it grows exponentially
in length.
Another quantity which exponentially grows is the ρ component of the
Rindler momentum. To see this, we use the transformation in equation
14.0.1 to derive
∂
∂
√1 −e−τ ∂ + − + eτ
− ,
∂ρ =
∂X
∂X
2
Strings
159
or in terms of momenta
1 Pρ = √ eτ P− − e−τ P+
2
(14.2.25)
In the Rindler approximation to a black hole horizon, P± are conserved,
and therefore as τ → ∞ the radial momentum Pρ grows like eτ . Evidently
then the ratio of the string length to its total radial momentum is fixed. As
the string falls toward the horizon, its radial momentum increases by the
mechanism of its physical length increasing.
14.3
Interactions
In Chapter 7 we saw that a charge falling toward the stretched horizon
spreads over an area which grows exponentially. The area occupied by a
free string only grows linearly. However, this not the end of the story. The
the total length of the string grows exponentially with τ . It is clear that
this behavior cannot continue indefinitely. The exponential growth of string
length and linear growth of area imply that the transverse density of string
increases to the point where string interactions must become important
and seriously modify the free string picture. √
Roughly speaking, when a
piece of string gets within a distance of order α of another piece, they
can interact. The number of such string encounters will obviously increase
without bound as τ → ∞.
String interactions are governed by a dimensionless coupling constant g
which determines the amplitude for strings to rearrange when they cross.
Obviously the importance of interactions is governed not only by g, but
also by the local density of string crossings. Let ρ be the number of such
crossings per unit horizon area. When g 2 ρ becomes large, interactions can
no longer be ignored.
Now, the form g 2 ρ is not dimensionless. There is only one dimensional
constant in string theory, the inverse string
√ tension α with units of area.
Sometimes α is replaced by a length s = α . The dimensionally correct
statement is that string interactions become important when
g2ρ ≥
1
2
s
≈
1
g 2 α
(14.3.26)
This criterion has a profound significance. The quantity g 2 α in string
theory also governs the gravitational interaction between masses. It is
160
Black Holes, Information, and the String Theory Revolution
none other than the gravitational coupling constant (in units with c =
= 1). The implication is that interactions become important when the
1
, the area density of horizon entropy.
area density of the string approaches G
Although we cannot follow the string past the point where interactions
become important, we can be sure that something new will happen. A
1
. Since the
good guess is that the density of string saturates at order G
τ
total length of string grows like ≈ e the area that it occupies must also
grow exponentially. This is reminiscent of the pattern of growth that we
encountered in Chapter 7.
Verifying that as δ → 0 the string grows as if the density saturates is
beyond the current technology of string theory. But the simple assumption
that splitting and joining interactions cause effective short range repulsion,
and that the repulsion prevents the density from increasing indefinitely,
provides a phenomenological description of how information spreads over
the horizon. Since the spreading is associated with a decreasing time of
averaging it is not seen by a freely falling observer. This is the essence of
complementarity.
In general, string theory is not a 4-dimensional theory. It is important
to check if the same logic applies in higher dimensions. Let D be the spacetime dimension. The general case goes as follows:
Since the number of transverse directions is D-2, equation 14.3.26 is replaced by
g2 ρ ≥
1
D−2
s
=
1
(α )
D−2
2
(14.3.27)
or
ρ ≈
1
g 2 D−2
s
(14.3.28)
In D dimensions, the gravitational and string couplings are related by
G ≈ g2
D−2
s
(14.3.29)
so that the perturbative limit is again reached when the density is of order
ρ ≈
1
G
(14.3.30)
Strings
14.4
161
Longitudinal Motion
In discussing the properties of horizons, we have repeatedly run into
the idea of the stretched horizon, located a microscopic distance above
the mathematical horizon. It is natural to ask if the stretched horizon
has any reality or whether it is just a mathematical fiction? Of course,
to a Frefo observer, neither the stretched nor the mathematical horizon
appears real. But to a Fido the stretched horizon is the layer containing
the physical degrees of freedom that give rise to the entropy of the horizon.
Thus, consistency requires that in some appropriate sense, the degrees of
freedom of an infalling object should get deposited in this layer of finite
thickness. To study this question we must examine how strings move in
the X − direction.
It is a curious property of string theory that X − (σ) is not an independent degree of freedom. The only degrees of freedom carried by the string
are the transverse coordinates X i (σ) which lie in the plane of the horizon.
The longitudinal location is defined by an equation whose origin is in the
gauge fixing to the light cone gauge. The derivation is provided in the
supplement that follows this discussion.
∂X i ∂X i
∂X −
=
∂σ1
∂σ1 ∂σ0
(14.4.31)
Recall that the quantum theory of X(σ) is a simple (1+1 dimensional)
quantum field theory of (D-2) free scalar fields defined on a unit circle. In
such a theory, local operators can be characterized by a mass dimension.
i
For example, X i has dimension zero, while ∂X
∂σ has dimension 1. The right
side of equation 14.4.31 has dimension 2. Thus it is apparent that X − (σ)
has dimension 1. It immediately follows that the fluctuation in X − satisfies
∆X − ∼
2
s
(14.4.32)
δ
The factor 2s is needed for dimensional reasons.
Another way to write equation 14.4.32 is to observe that δ is an averaging time in light cone coordinates. In other words δ = ∆X + . Equation
14.4.32 then takes the form of an uncertainty principle
∆X − ∆X + ≈
2
s.
(14.4.33)
Now the geometric meaning of equation 14.4.33 is very interesting. Let us
draw the motion of the fluctuating string in the X ± plane as it falls towards
162
Black Holes, Information, and the String Theory Revolution
∆x
-
∆τ}
x+
Region of
increasing
fluctuation
Fig. 14.2
Near horizon Rindler time slices
the horizon. Evidently as X + tends to zero the fluctuation in X − required
by equation 14.4.33 must increase. This is shown in Figure 14.2. What the
figure illustrates is that the stringy material tends to fill a region out to a
fixed proper distance from the mathematical horizon at X + = 0. In other
words, unlike a point particle, the stringy substance is seen by a probe to
hover at a distance ∼ s above the horizon. Once again this surprising result
is a direct consequence of arbitrarily high frequency fluctuations implicit
in the stringy structure of matter. Note that if the string coupling satisfies
gs << 1, the string length s can be much greater than the Planck length,
P = s g. In this case it is the string length and not the Planck length
that controls the distance of the stretched horizon from the mathematical
horizon.
Strings
163
Supplement: Light cone gauge fixing of longitudinal string
motions
The coordinate X− (τ ,σ) is not an independent degree of freedom, but is
+
given by equation 14.4.31. To see this multiply 14.4.31 by 1 = ∂X
∂τ , which
is true for light cone coordinates. Thus we must check the validity of the
following equation
∂X j ∂X j
∂X + ∂X −
−
= 0.
∂τ ∂σ
∂τ ∂σ
(14.4.34)
The content of this equation expresses the underlying invariance of string
theory to a reparameterization of the σ coordinate. Under the transformation
σ → σ + δσ
the X s transform as
X→X+
∂X
δσ.
∂σ
(14.4.35)
The Noether charge for this invariance is exactly the quantity in equation
14.4.34. Setting this quantity to zero insures that the physical spectrum
consists only of states which are invariant under σ reparameterization. In
going to the light cone frame the constraint serves to define X − in terms of
the transverse coordinates. However even in the light cone gauge there is
a bit of residual gauge invariance, namely shifting σ by a constant. In the
light cone frame only the transverse X s are dynamical and the generator
of these rigid shifts is
σ’=0
vs
σ=0
Fig. 14.3
String parameter translational invariance
164
Black Holes, Information, and the String Theory Revolution
dσ
∂X j dX j
∂σ ∂τ
which by equation 14.4.31 is equal to
/
∂X −
dσ
.
∂σ
(14.4.36)
(14.4.37)
Setting this to zero simultaneously insures invariance under shifts of σ and
periodicity of X − (σ).
Chapter 15
Entropy of Strings and Black Holes
The Bekenstein–Hawking entropy of black holes points to some kind of
microphysical degrees of freedom, but it doesn’t tell us what they are. A
real theory of quantum gravity should tell us and also allow us to compute the entropy by quantum statistical mechanics, that is, counting microstates. In this lecture we will see to what extent string theory provides
the microstructure and to what extent it enables us to compute black hole
entropy microscopically.
String theory has many different kinds of black holes, some in 3 + 1
dimensions, some in higher dimensions. The black holes can be neutral or
be charged with the various charges that string theory permits. We will see
that for the entire range of such black holes, the statistical mechanics of
strings allows us to compute the entropy up to numerical factors of order
unity. In every case the results nontrivially agree with the Bekenstein–
Hawking formula. What is more, in one or two cases in which the black
holes are invariant under a large amount of supersymmetry the calculations
can be refined and give the exact numerical coefficients. All of this is in
cases where quantum field theory would give an infinite result.
Because string theory is not necessarily a 4-dimensional theory, it is
worth exploring the connections between strings and black holes in any
dimension. Let us begin with the formula for the entropy of a Schwarzschild
black hole in an arbitrary number of dimensions. Call the number of spacetime dimensions D. The black hole metric found by solving Einstein’s
equation in D dimensions is given by
dτ 2 =
RSD−3
1 − D−3
r
dt2 −
RSD−3
1 − D−3
r
165
−1
dr2 − r2 dωD−2
(15.0.1)
166
Black Holes, Information, and the String Theory Revolution
The horizon is defined by
RS =
16π(D − 3)GM
ΩD−2 (D − 2)
1
D−3
(15.0.2)
and its D-2 dimensional “area” is given by
A = RSD−2
dΩD−2 = RSD−2 ΩD−2 .
(15.0.3)
Finally, the entropy is given by
D−2
S =
(2GM ) D−3 ΩD−2
A
=
.
4G
4G
(15.0.4)
The entropy in equation 15.0.4 is what is required by black hole thermodynamics.
Supplement: Schwarzschild geometry in D = d + 1
dimensions
To extend a static spherically symmetric geometry to D = d + 1 dimensions, the metric can be assumed to be of the form
ds2 = −e2Φdt2 + e2∆ dr2
(15.0.5)
2
.
+ r2 dθ12 + sin2 θ1 dθ22 + ... + sin2 θ1 ...sin2 θd−2 dθd−1
Using orthonormal coordinates, the Gt̂t̂ component of the Einstein tensor
can be directly calculated to be of the form
(D − 2)(D − 3)
e−2∆
+
(1 − e−2∆ )
(15.0.6)
Gt̂t̂ = − (D − 2)∆
r
2r2
From Einstein’s equation for ideal pressureless matter, Gt̂t̂ = κρ. This
means
Gt̂t̂ = −
(D − 2) d D−3
r
(1 − e−2∆ ) = −κρ
D−2
2r
dr
(15.0.7)
which can be solved to give
(1 − e−2∆ )rD−3 =
2κ
D−2
r
0
ρ(r )r
D−2
dr =
M
2κ
(D − 2) ΩD−2
(15.0.8)
Entropy of Strings and Black Holes
167
(D−1)/2
2π
where the solid angle is given by ΩD−2 = Γ((D−1)/2)
. The Gr̂r̂ component
of the Einstein tensor satisfies
−2∆
−2∆
(1
−
e
)
Gr̂r̂ = − −(D − 2)Φ e r + (D−2)(D−3)
2
2r
(15.0.9)
e−2∆
= − −(D − 2)(Φ + ∆ ) r + κρ
For pressureless matter in the exterior region (ρ = 0 = P ), we can immediately conclude that Φ = −∆. Defining the Schwarzschild radius
2κM
we obtain the form of the metric
RSD−3 = (D−2)Ω
D−2
e−2∆ = 1 −
RS
r
D−3
= e2Φ
(15.0.10)
If we write F (r) ≡ e2Φ , a useful shortcut for calculating the solution to Einstein’s equation 15.0.7 is to note its equivalence to the Newtonian Poisson
equation in the exterior region
∇2 F (r) = −κρ ,
F (r) = 1 + 2φN ewton .
(15.0.11)
The Hawking temperature can be calculated by determining the dimensional factor between the Rindler time and Schwarzschild time. Near the
horizon, the proper distance to the horizon is given by
D−3
r
2RS
−1
(15.0.12)
ρ =
(D − 3)
Rs
which gives the relation between Rindler time/temperature units and
Schwarzschild time/temperature units
dω =
(D − 3)
dt
2RS
(15.0.13)
Thus, the Hawking temperature of the black hole is given by
THawking =
1 (D − 3)
.
2π 2RS
(15.0.14)
Using the first law of thermodynamics, the entropy can be directly calculated to be of the form
S =
2π(D − 3)A
κ
(15.0.15)
Substituting the form κ = 8π(D − 3)G for the gravitational coupling gives
the previous results in D-dimensions.
168
Black Holes, Information, and the String Theory Revolution
All calculations of entropy in string theory make use of a well known
trick of quantum mechanics. The trick consists of identifying some kind
of control parameter that can be adiabtically varied. In the process of
adiabatic variation, energy levels are neither created nor destroyed. Thus
if we can follow the system to a value of the control parameter where the
system is tractable we can count the states easily even if the nature of the
object changes during the variation. Basically we are using the quantum
analog of the method of adiabatic invariants.
The trick in the string theory context is to vary the strength of the string
coupling adiabatically until we arrive at a point where the gravitational
forces are so weak that the black hole “morphs” into some more tractable
object. Thus we begin with a black hole of mass Mo in a theory with string
coupling go . Adiabatically varying a control parameter like go will cause a
change in the black hole mass and other internal structural features. But
such a variation will not alter its entropy. Entropy is an adiabatic invariant.
Let us imagine decreasing the string coupling g. What happens to the
black hole as g tends to zero? The answer is obvious. It must turn into
a collection of free strings. String theory has all kinds of non-perturbative
objects, branes of various dimensionality such as membranes, D-branes,
monopoles, and so on. But only the free strings have finite energy in the
limit g → 0. Therefore a neutral black hole must evolve into a collection of
free strings. A very massive black hole might evolve into a large number of
low mass strings or, at the opposite extreme, a single very highly excited
string.
Very highly excited free strings have an enormously rich spectrum. They
can be thought of as a mass of tangled string that forms a time-varying
random walk in space. Such random walking strings have a large entropy
and can be studied statistically.
The entropy of a string of mass m can be calculated by returning to the
light cone quantization of the previous lecture. For any eigenstate of the
Hamiltonian with vanishing transverse momentum and unit P− the light
cone energy is m2 /2.
On the other hand the quantization of the string defines a 1+1 dimensional quantum field theory in which the (D-2) transverse coordinates X i (σ)
play the role of free scalar fields. The spatial coordinate of this field theory
is σ1 , and it runs from 0 to 2π.
The counting of the states of a free string is best done in the light
Entropy of Strings and Black Holes
169
cone version of the theory that we discussed in the last lecture. In order to
describe the highly excited string spectrum, a formal light cone temperature
T can be defined. Recall that the free string is described by means of a
1 + 1 dimensional quantum theory containing D − 2 fields X i .
The entropy and energy of such a quantum field theory can be calculated
by standard means. The leading contribution for large energy is (setting
s = 1)
E = π T 2 (D − 2)
(15.0.16)
S = 2πT (D − 2)
2
Using E = m2 and eliminating the temperature yields S =
or, restoring the units
S = 2(D − 2)π m s .
Subleading corrections can also be calculated to give
S = 2(D − 2)π m s − c log (m s )
2(D − 2)π m
(15.0.17)
(15.0.18)
where c is a positive constant. The entropy is the log of the density of
states. Therefore the number of states with mass m is
c
1
(15.0.19)
exp
2π(D − 2) m s
Nm =
m s
The formula 15.0.19 is correct for the simplest bosonic string, but similar
formulae exist for the various versions of superstring theory.
Now let us compare the entropy of the single string with that of n
strings, each carrying mass m
n . Call this entropy Sn (m). Then
Sn (m) = n S(m/n)
or
Sn (m) =
2(D − 2)π m
(15.0.20)
s
− n c log
m s
n
(15.0.21)
Obviously for large n the single string is favored. This is actually quite
general. For a given total mass, the statistically most likely state in free
string theory is a single excited string. Thus it is expected that when the
string coupling goes to zero, most of the black hole states will evolve into
a single excited string.
170
Black Holes, Information, and the String Theory Revolution
These observations allow us to estimate the entropy of a black hole. The
assumptions are the following:
• A black hole evolves into a single string in the limit g → 0
• Adiabatically sending g to zero is an isentropic process; the entropy of
the final string is the same as that of the black hole
• The entropy of a highly excited string of mass m is of order
S ∼ m
(15.0.22)
s
• At some point as g → 0 the black hole will make a transition to a string.
The point at which this happens is when the horizon radius is of the
order of the string scale.
To understand this last assumption begin with a massive black hole.
Gravity is clearly important and cannot be ignored. But no matter how
massive the black hole is, as we decrease g a point will come where the
gravitational constant is too weak to matter. That is the point where the
black hole makes a transition and begins to act like a string.
The string and Planck length scales are related by
g2
D−2
s
=
D−2
.
p
(15.0.23)
Evidently as g decreases the string length scale becomes increasingly big in
Planck units. Eventually, at some value of the coupling that depends on
the mass of the black hole, the string length will exceed the Schwarzschild
radius of the black hole. This is the point at which the transition from black
hole to string occurs. In what follows we will vary the g while keeping fixed
the string length s . This implies that the Planck length varies.
Let us begin with a black hole of mass Mo in a string theory with
coupling constant go . The Schwarzschild radius is of order
1
RS ∼ (Mo G) D−3 ,
(15.0.24)
and using
G ≈ g2
D−2
s
(15.0.25)
we find
RS
s
≈
s
Mo go2
1
D−3
.
(15.0.26)
Entropy of Strings and Black Holes
171
Thus for fixed go if the mass is large enough, the horizon radius will be
much bigger than s .
Now start to decrease g. In general the mass will vary during an adiabatic process. Let us call the g-dependent mass M (g). Note
M (go ) = Mo
(15.0.27)
The entropy of a Schwarzschild black hole (in any dimension) is a function of the dimensionless variable M P . Thus, as long as the system remains a black hole,
M (g)
Since
P
≈
s
g
2
D−2
P
= constant.
(15.0.28)
we can write equation 15.0.28 as
M (g) = Mo
go2
g2
1
D−2
.
(15.0.29)
Now as g → 0 the ratio of the g-dependent horizon radius to the string
scale decreases. From equation 15.0.2 it becomes of order unity at
M (g)
D−2
P
≈
D−3
s
(15.0.30)
which can be written
M (g)
s
≈
1
.
g2
(15.0.31)
Combining equations 15.0.29 and 15.0.31 we find
1
D−2
M (g)
s
≈ MoD−3 GoD−3 .
(15.0.32)
As we continue to decrease the coupling, the weakly coupled string
mass will not change significantly. Thus we see that a black hole of mass
Mo will evolve into a free string satisfying equation 15.0.32. But now we
can compute the entropy of the free string. From equation 15.0.22 we find
D−2
1
S ≈ MoD−3 GoD−3 .
(15.0.33)
This is a very pleasing result in that it agrees with the Bekenstein–Hawking
entropy in equation 15.0.4. However, in this calculation the entropy is
calculated as the microscopic entropy of fundamental strings.
The evolution from black hole to string can be pictorially represented
by starting with a large black hole. The stretched horizon is composed of a
172
Black Holes, Information, and the String Theory Revolution
il d
ch
rzs
wa
ch
S
R
S
Evolution from black hole to string. (a) A black hole with stringy
stretched horizon smaller than Schwarzschild radius, (b) with stretched horizon
and string scale comparable to radius scale, and (c) turned into a string
Fig. 15.1
stringy mass to a depth of ρ = s as in the diagram Figure 15.1a. The area
1
. Another important property of the
density of string is saturated at ∼ G
stretched horizon is its proper temperature. Since the proper temperature
1
, the temperature of the stringy mass will be
of a Rindler horizon is 2πρ
TStretched ≈
1
s
This temperature is close to the so-called Hagedorn temperature, the maximum temperature that a string can achieve.
As the Schwarzschild radius is decreased (in string units), the area of
the horizon decreases but the depth of the stretched horizon stays fixed
as in Figure 15.1b. Finally the horizon radius is no larger than s (Figure
15.1c) and the black hole turns into a string.
By now a wide variety of black holes that occur in string theory have
been analyzed in this manner. The method is always the same. We adiabatically allow g to go to zero and identify the appropriate string configuration
that the black hole evolves into.
A particularly interesting situation is that of charged extremal black
holes which may be supersymmetric configurations of a supersymmetric
theory. In this case the extremal black hole is absolutely stable and in
addition, its mass is completely determined by supersymmetry. When this
occurs there is no need to follow the mass of the black hole as g varies;
the mass is fixed. Under these conditions the black hole can be compared
Entropy of Strings and Black Holes
173
directly to the corresponding weakly coupled string configuration and the
entropy read off from the degeneracy of the string theory spectrum. In the
cases where exact calculations are possible the charges carried by black holes
are associated not with fundamental strings but D-branes. Nevertheless the
principles are that same as those that we used to study the Schwarzschild
black hole in D dimensions. The results in these more complicated examples
are in precise agreement with the Hawking–Bekenstein entropy.
Hagedorn Temperature Supplement
On general grounds, one can determine the density of states η for the
various string modes m:
√
η(m) ∼ exp(4πm α )
This allows the partition function to be written as
Z∼
∞
0
√
exp(4πm α )exp − m
T dm
which diverges if the temperature T is greater than the Hagedorn temperature defined by
THagedorn ≡
1
√
4π α
The Hagedorn temperature scales with the inverse string length
THagedorn ∼
1
ls .
To get a feel for the scale of the Hagedorn temperature, recall the behavior of the entropy given by S∼log(density of states). Using dimensional
considerations, we have seen that the entropy of the string scales like
Sstring ∼ df Ms ls ,
where d f is the number of internal degrees of freedom available. Thus, the
density of states behaves like
√
eSstring ∼ e df Ms ls ∼ e1/THagedorn
which gives the scale T Hagedorn ∼ 1/l s . If one examines multi-string fluctuations as a function of temperature, the Hagedorn temperature is the
174
Black Holes, Information, and the String Theory Revolution
“percolation” temperature for multiple strings fluctuations to coalesce into
fluctuations of a single string as represented in Figure 15.2.
Increase
Temp
Increase
Temp
T<TH
T=TH
Fig. 15.2
String “percolation”
T>TH
Conclusions
The views of space and time that held sway during most of the 20th century
were based on locality and field theory, first classical field theory and later
quantum field theory. The most fundamental object was the space-time
point or better yet, the event. Although quantum mechanics made the
event probabilistic and relativity made simultaneity non-absolute, it was
assumed that all observers would agree on the usual invariant relationships
between events. This view persisted even in classical general relativity. But
the paradigm is gradually shifting. It was never adequate to deal with the
combination of quantum mechanics and general relativity.
The first sign of this was the failure of standard quantum field theory methods when applied to the Einstein action. For a long time it was
assumed that this just meant that the theory was incomplete at short distances in the same way that the Fermi theory of weak interactions was
incomplete. But the dilemma of apparent information loss in black hole
physics that was uncovered by Hawking in 1976 said otherwise. In order
to reconcile the equivalence principle with the rules of quantum mechanics
the rules of locality have to be massively modified. The problem is not a
pure ultraviolet problem but an unprecedented mix of short distance and
long distance physics. Radical changes are called for.
The new paradigm that is gradually emerging is based on four closely
related concepts. The first is Black Hole Complementarity. This principle is
a new kind of relativity in which the location of phenomena depends on the
resolution time available to the experimenter who probes the system. An
extreme example would be the fate of someone, call her Alice, falling into an
enormous black hole with Schwarzschild radius of a billion years. According
to the low frequency observer, namely Alice herself, or someone falling with
her, nothing special is felt at the horizon. The horizon is harmless and she
175
176
Black Holes, Information, and the String Theory Revolution
or her descendants can live for a billion years before being crushed at the
singularity.
In apparent complete contradiction, the high frequency observer who
stays outside the black hole finds that his description involves Alice falling
into a hellish region of extreme temperature, being thermalized, and eventually re-emitted as Hawking radiation. All of this takes place just outside
the mathematical horizon. Obviously this has to do with more than just a
modification of the short distance physics. As we have seen, the key to black
hole complementarity is the extreme red shift of the quantum fluctuations
as seen by the external observer.
The second new idea is the Infrared/Ultraviolet connection. Very closely
related to Black Hole Complementarity, the IR/UV connection reverses one
of the most fundamental trends of 20th century physics. Throughout that
century a close connection between energy and size prevailed. If one wished
to study progressively smaller and smaller objects one had to use higher and
higher energy probes. But once gravity is involved that trend is reversed.
At energies above the Planck scale any possible short distance physics that
we might look for is shrouded behind a black hole horizon. As we raise the
energy we wind up probing larger and larger distance scales. The ultimate
implications of this, especially for cosmology are undoubtedly profound but
still unknown.
Third is the Holographic Principle. In many ways this is the most
surprising ingredient. The non-redundant degrees of freedom that describe
a region of space are in some sense on its boundary, not its interior as they
would be in field theory. At one per Planck area, there are vastly fewer
degrees of freedom than in a field theory, cutoff at the Planck volume. The
number of degrees of freedom per unit volume becomes arbitrarily small as
the volume gets large. Although the Holographic Principle was regarded
with skepticism at first it is now part of the mainstream due to Maldacena’s
AdS/CFT duality. In this framework the Holographic Principle, Black
Hole Complementarity and the IR/UV connection are completely manifest.
What is less clear is the dictionary for decoding the CFT hologram.
Finally, the existence of black hole entropy indicates the existence of
microscopic degrees of freedom which are not present in the usual Einstein theory of gravity. It does not tell us what they are. String theory
does provide a microscopic framework for the use of statistical mechanics.
In all cases the entropy of the appropriate string system agrees with the
Bekenstein–Hawking entropy. This, if nothing else, provides an existence
proof for a consistent microscopic theory of black hole entropy.
Conclusions
177
The theory of black hole entropy is incomplete. In each case a trick,
specific to the particular kind of black object under study, is used to determine the relation between entropy and mass for the specific string-theoretic
object that is believed to represent a particular black hole. Then classical general relativity is used to determine the area–mass relation and the
Bekenstein–Hawking entropy. In no case do we use string theory directly
to compare entropy and area. In this sense the complete universality of the
area–entropy relation is still not fully understood.
One very large hole in our understanding of black holes is how to think
about the observer who falls through the horizon. Is this important? It is
if you are that observer. And in some ways, an observer in a cosmological
setting is very much like one behind a horizon. At the time of the writing
of this book there are no good ideas about the quantum world behind the
horizon. Nor for that matter is there any good idea of how to connect the
new paradigm of quantum gravity to cosmology. Hopefully our next book
will have more to say about this.
178
Black Holes, Information, and the String Theory Revolution
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1. Kip
S.
Thorne,
Richard
H.
Price
and
Douglas
A.
MacDonald.
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2. Don N. Page. Information in Black Hole Radiation, hep-th/9306083, Phys.
Rev. Lett. 71 (1993) 3743–3746.
3. Leonard Susskind. The World as a Hologram, hep-th/9409089, J. Math. Phys.
36 (1995) 6377–6396.
4. S. Corley and T. Jacobson. Focusing and the Holographic Hypothesis, grqc/9602043, Phys. Rev. D53 (1996) 6720–6724.
5. W. Fischler and L. Susskind. Holography and Cosmology, hep-th/9806039.
6. Raphael Bousso. The Holographic Principle, hep-th/0203101, Rev. Mod.
Phys. 74 (2002) 825–874.
7. Juan M. Maldacena. The Large N Limit of Superconformal Field Theories
and Supergravity, hep-th/9711200, Adv. Theor. Math. Phys. 2 (1998) 231–
252; Int. J. Theor. Phys. 38 (1999) 1113–1133.
8. Edward Witten. Anti De Sitter Space and Holography, hep-th/9802150, Adv.
Theor. Math. Phys. 2 (1998) 253–291.
9. L. Susskind and Edward Witten. The Holographic Bound in Anti-de Sitter
Space, hep-th/9805114.
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179
180
Black Holes, Information, and the String Theory Revolution
Index
AdS(5) ⊗ S(5), 128, 141
’t Hooft limit, 135
D-branes, 136
de Sitter space, 119
degrees of freedom, 138
density matrix, 34, 45, 71
adiabatic invariants, 168
adiabatic variation (string coupling),
170
AdS black hole, 144
AdS/CFT correspondence, 133
angular momentum, 27
anti de Sitter space, 123, 128
effective potential, 26
electrical properties, 62
electromagnetic field, 62
electrostatics, 63
energy (horizon), 52
entanglement, 32, 71, 85
entanglement entropy (equality), 72
entropy, 52, 61, 102, 165
entropy (Bekenstein–Hawking), 51,
165
entropy (bounds), 101
entropy (calculation), 36, 43, 168
entropy (coarse grained), 73, 76
entropy (fine grained), 70, 73
entropy (infinite), 81
entropy (maximum), 102
entropy (quantum field theory), 81
entropy (strings), 165, 168
entropy (thermal), 35, 46, 73
entropy (vacuum), 45
entropy (Von Neumann), 35, 70, 76
entropy of entanglement, 35, 46, 71
equivalence principle, 21, 31, 69, 77
evaporation, 48
evaporation time, 54
expansion rate, 113
extended horizon, 11
barrier penetration, 28
baryon number violation, 89
Birkoff’s theorem, 17
black hole formation, 15
Boltzmann factor, 40
Bousso’s construction, 114, 120, 121
brick wall, 84
caustic lines, 106
charge, 55
charged black holes, 55
classical fields, 31
coarse graining, 69
collapsing light-like shell, 103
complementarity, 85, 97, 157, 160
conductivity, 68
conformal field theory, 131
conformally flat, 7
cosmological constant, 119
curvature, 5
181
182
Black Holes, Information, and the String Theory Revolution
extremal black hole, 56
luminosity, 54
Feynman–Hellman theorem, 37
Fidos, 21, 32, 39, 41, 48, 58, 161
fiducial observer (see also Fidos), 21
Fischler–Susskind bound, 110
fluctuations, 41, 60, 93
focusing theorem, 107
free field approximation, 49
freely falling observer (see also
Frefos), 22
Frefos, 21, 41, 85
Friedman–Robertson–Walker
geometry, 110
Minkowski space, 9, 106
mirror boundary condition, 44
momentum, 23, 32
gauge fixing, 22
geodesic completeness, 129
ground state (charged black hole), 59
Hagedorn temperature, 172
Hawking, 81
Hawking radiation, 49, 85
high frequency phenomena, 151
holographic principle, 101, 127, 130
holography, 101, 127
holography (AdS space), 130
horizon, 4, 8, 20, 25, 44, 52, 57, 144,
152
hyperbolic plane, 130
infalling observer, 5
inflation, 121
information, 74, 97, 144
information conservation, 69, 81
information retention time, 77
Kruskal–Szekeres coordinates, 10
lattice of discrete spins, 101
laws of nature, 69
level density, 52
light cone gauge fixing, 163
light cone quantum mechanics, 153
light cone string theory, 156
Liouville’s theorem, 69
longitudinal string motions, 161
near horizon coordinates, 8
near horizon wave equation, 28
Newton’s constant, 134
no-cloning principle, 79
pair production, 55
particle horizon, 112
path integral, 37
Penrose diagram (AdS black hole),
142
Penrose diagram (de Sitter space),
120
Penrose diagram (F.R.W. space), 115
Penrose diagrams, 14
Penrose–Bousso diagram, 117
Penrose–Bousso diagram (AdS), 123
Planck distance, 23
Planck length, 52, 62, 134, 170
Planck units, 127
Poincaré disk, 130
proper distance, 8, 167
proton decay, 89
quantization rules, 44
quantum field theory, 43
quantum fields, 25, 61, 102, 151
quantum fields (Rindler space), 31
quantum Xerox principle, 69, 79
red shift, 48
reheating, 121
Reissner–Nordstrom black hole, 55
resistance, 65
Rindler energy, 53
Rindler Hamiltonian, 32, 44
Rindler space, 8, 31, 152, 167
S-matrix, 81
scalar wave equation, 25
Schwarzschild black hole, 3
Index
Schwarzschild coordinates, 3
Schwarzschild geometry
(D-dimensions), 166
second law of thermodynamics, 103
solid angle, 167
standard thermometer, 39
static observers, 21
stretched horizon, 42, 61, 62, 161
string coupling constant, 134
string interactions, 159
string percolation, 174
strings, 151
Supersymmetric Yang–Mills (SYM),
133
supersymmetry, 131, 133
surface charge density, 63
temperature (Hawking), 48, 167
temperature (proper), 39
temperature (Rindler), 39
thermal atmosphere, 45, 48
thermal ensemble, 39
thermal fluctuations, 144
thermodynamic instability, 141
thermodynamics, 51
tidal forces, 5
tortoise coordinates, 3, 7, 65
tortoise-like coordinates, 28
transfer matrix, 38
transverse spreading, 155
Unruh, 36, 39
UV/IR connection, 95, 101, 135
vacuum, 45
183